Term Paper: Econ 290Global Imbalances and Exchange Rate Exposure

William Swanson

Abstract

This paper extends the framework of Gabaix and Maggiori (2013) to show that the netshare of liabilities in domestic currency is relevant in pricing the riskiness of a countriescurrency. I develop several new risk factors that can theoretically explain part of thecurrency excess returns from a Carry Trade-type strategy. Building on the empiricalwork of Della Corte, Riddiough and Sarno (2013) , I test the risk factors across a set offive currency portfolios and check the results for robustness. While some factors price riskas well as the High-Low Factor,1 none are better and the results are varied.

The connection between exchange rate fluctuations and imbalances in trade and capi-

tal flows have received a great deal of recent attention among economists. In the lead up

to the financial crisis and in the fall out, large imbalances in net asset positions sparked

protracted academic debate and empirical work on net positions2. Because the crisis was

coupled with sharply falling asset prices, it is natural to look to global imbalances to aid

in understanding the global asset pricing dynamics. Theoretically, exchange rate fluctu-

ations, currency risk premia and global imbalances are linked. The empirical literature

has made some recent success in characterizing the linkages. The work by Della Corte,

Riddiough and Sarno (2013) , hereafter referred to as DRS, argues convincingly that a

countries external debt determines if the local currency appreciates or depreciates in times

of crisis. The real world provides evidence. Brazil, South Africa, and Turkey all expe-

rienced sharp depreciation in 2012, and all these countries have something in common:

they are large external debtors. If foreign debts imply currency risk, then a simple asset

pricing model would predict that the exchange rate depreciates in times of global crisis.

The theory is not full-proof, however. There are other big debtor nations, most notably

the U.S., that enjoy exchange rate appreciations during times of crisis and higher aver-

age returns throughout the business cycle, which are well documented phenomena via in

Gourinchas and Rey (2007) and Gourinchas, Rey and Govillot (2010) . This paper will

1The High-Low factor is the slope factor discussed in DRS and derived in Lustig et. al. (2011). The factoris equivalent to a Carry strategy. See the Literature Review section of this paper for details.

2See Caballero et. al. (2008) for a good discussion about the context of global imbalances.

1

adapt the DRS empirical approach to better account for these observations with an eye

towards a comprehensive understanding of exchange rate fluctuations and currency risk

premia.

The framework developed by Gabaix and Maggiori (2014) , hereafter referred to as

simply GM, is useful for understanding how imbalances influence currency premia. In

their imperfect markets, two country framework, a financier can hedge risk between coun-

tries by taking long and short positions in each country’s bonds. Countries can hold

stocks of debt, import and export goods, and hedge risk next period with one-period non-

contingent bonds. The financier provides insurance against next periods consumption risk

at the price of a risk premium in the form of a depreciated (appreciated) exchange rate

for currencies in which the financier is short (long). The GM model, then, provides a the-

oretical explanation for positive excess returns to the Carry Trade (in which an investor

goes long in high interest rate currencies and short in low interest rate currencies). DRS

tests this theory directly using a linear factor pricing model to show that foreign asset

imbalances can partly explain the excess returns on currency portfolios. Specifically, the

GM model predicts that debtor countries with high shares of debt denominated in foreign

currency should expect a currency depreciation in a crisis. Conversely, creditor countries

should experience an appreciation in times of crisis. While much of the U.S. foreign debt

is denominated in U.S. dollars (on average 82% since 1990), the U.S. Net Foreign Asset

position is negative (liabilities are roughly 25% greater than assets), and DRS acknowl-

edge that countries like the U.S. are a notable exception to the GM theory. With a simple

amendment to the GM framework, I show that the net share, not just the share, of li-

abilities in foreign currency is the relevant metric. I also extend the GM framework in

several ways to show that other risk factors might be useful in pricing currency risk. For

example, I develop a risk factor that is approximately a measure of the foreign exchange

rate exposure to the Net Foreign Asset Position. Additionally, I improve on the standard

momentum strategy of currency returns by adjusting a standard momentum strategy by

a countries NFA position. With the new risk factors in hand, I closely follow the DRS

empirical framework and the robustness tests of Burnside (2011). Using data on currency

spot rates and forward rates, I replicate the empirical results of DRS and test my new

risk factors.

The paper proceed as follows. First I explain the literature in the variety of topics

2

to which this paper is related. Then I explain the theoretical framework of GM and I

develop several extensions. Following the theory, I define the risk factors I will test and

the empirical framework in which I will test them. I conclude with a discussion of the

results.

1 Literature

From the literature on wealth effects, Gourinchas and Rey (2005) show that a depreciation

of the U.S. dollar has two notable impacts: the trade adjustment channel, where exports

become cheaper, and the valuation channel, where foreign assets appreciate more than

foreign liabilities when a smaller share of foreign assets are denominated in a foreign

currency. This valuation effect can be substantial, accounting for up to 10% of GDP in

1994 and on that order in the last two decades. Most studies on asset prices ignore the

valuation effect when accounting for the earnings and wealth of a U.S. investor. This is

surprising given that the valuation channel is responsible for up to 30% of international

current account adjustment and is operative in the short and medium term. These facts

imply that the stock and denomination of external debt is vital for determining the wealth

of the average investor and the medium to long-term exchange rate movements.

Gourinchas (2008) develops formal intuition around the causes and consequences of

valuation effects. Using an inter-temporal approach to the current account, the paper

argues that the NFA position is the relevant metric for foreign exposure because it in-

cludes realized capital gains and losses arising from local currency movements and asset

price changes that are omitted from current account measures. The authors show that

accounting for the valuation component in the net imbalances is vital to calculating a

steady state real exchange rate and sustainable level of debt in the long run.

Phylaktix (2004), Dumas and Solnik (1995) and De Santis and Gerard (1998) find

strong support for the specification of the CAPM model when it includes controls for

both equity market and currency risk. Foreign currency risk is shown to be relevant to

investors, however, the authors provide no theoretical argument to explain how debt and

equity stocks should be considered.

This paper assumes that agents cannot hedge away all real-exchange rate risk, an as-

sumption for which there is broad support. Hau and Rey (2006) estimate that only 10

percent of foreign equity positions are hedged, often due to institutional restrictions on

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the use of derivative contracts. Levich and Thomas (1993) calculated that forex risk hedg-

ing concerned only 8 percent of the total foreign equity investment. Portfolio managers

cited monitoring problems, lack of knowledge and public and regulatory perceptions as

most important reasons for the restricted forex derivative use. The development of the

derivative market notwithstanding, only a minor proportion of the total macroeconomic

forex return risk seems to be separately traded and eliminated. The typical foreign equity

investor holds currency return and foreign equity return risk as a bundle. Also, lack of

data means that the extent of cross-border currency hedging is difficult to assess; while the

volume of currency-related derivative trade is very large, much of this is between domestic

residents, which does not alter the aggregate net exposure of the economy. Furthermore,

if the counterparty in a derivatives contract is another domestic resident, the currency

risk still resides within the same country. Overall, the data indicate that most investor do

not completely hedge against all real exchange rate risk, and even if they did, economists

could not verify it easily.

Data Lane and Milesi-Ferretti, (2001), provided a set of useful annual estimates of net

and gross international investment positions for a sample of sixty-seven industrial and

developing countries. Their database covers the period from 1970 to 1998, thus adding at

least ten years of data to the IMFs IIP database. To construct net investment positions at

market value, Lane and Milesi-Ferretti devised ways to estimate the valuation component

of net imbalances from balance-of-payments (flows) data.

Data on external positions are again updated in Lane and Milesi-Ferretti (2004, 2007

), Lane and Shambaugh (2010) and finally updated to the period from 1990 to 2012 by

Benetrix, Lane and Shambaugh (2014). While the basic data set is available for the

entire 22 year period, the asset weighted exchange-rate indexes calculated in Lane and

Shambaugh (2010) are not updated beyond 2005.

Carry Returns The Uncovered Interest Parity (UIP) condition says that the differ-

ence between the home nominal interest rate and a foreign nominal interest rate should

equal the expected depreciation of the home currency against the foreign currency. This

condition rarely holds in the data (as first observed by Bilson (1981) and Fama (1984)).

Without parity, investors earn positive excess returns by investing in the foreign currency

with the highest interest rate, and funding the purchase with low interest rate curren-

4

cies, a strategy known as the Carry Trade. Among the explanations for why the parity

fails are: peso problems (e.g. Burnside et. al. (2006,2011); time varying risk premia

(e.g. Lustig and Verdelhan (2007, 2011)); long run productivity risk (e.g. Croce (2014));

infrequent or sticky portfolio decisions (e.g. Bacchetta and Van Wincoop (2010)); habit-

persistent preferences (e.g. Verdelhan 2010); and heterogeneous or incorrect information

(e.g. Bacchetta and Van Wincoop (2006)).

This paper builds from the literature on time varying currency risk premia as the

cause of the UIP failure. The argument goes: when domestic consumption growth is

low, high interest rate currencies tend to depreciate and low interest rate currencies tend

to appreciate. Therefore the Carry Trade is compensation for this expected deprecia-

tion. Additionally, if portfolio choices are made only infrequently (e.g. Kray and Ventura

(2003), Bacchetta and Van-Wincoop (2009) , Hau and Rey (2009) ), we must also ac-

count for endogenous risk factor, since the investor’s Stochastic Discount Factor (SDF)

is especially sensitive to currencies in which the investor is already heavily invested, that

is, the high interest rate ones. Likewise, the investor’s existing wealth and therefore the

SDF will be less sensitive to low interest rate currencies. Similar to the papers in this

literature, I show that the foreign currency composition of foreign assets is important to

an investors time-varying exposure to risk.

Yogo (2006) derives a linear factor pricing model in the style of Fama and Macbeth

(1991) from a consumer with Epstein Zin (1989, 1991) preferences. Also, in Yogo (2006)

the consumers have a running stock of durable goods, risk-less bonds and non-durable

consumption. This model sparked a literature of quarterly three-factor models that have

had varying success at explaining Carry Trade returns.

Using the framework developed in Yogo (2006), Lustig and Verdelhan (2007) sort

portfolios of foreign currency holdings by the foreign interest rate and argue that the

Carry Trade compensates consumers for short and long term consumption risk. Using

a linear factor pricing model, they find that most of the excess return are explained by

the comovements of non-durable consumption, durable consumption, and average U.S.

equity market returns. Burnside (2011) deflates these results by observing that much of

the models explanatory power is driven by (1) a large and statistically significant constant

that represents pricing error, and (2) ignored sampling uncertainty in the two-step Fama-

Macbeth procedure. In the empirical portion of this paper I take into careful consideration

5

the critiques of Burnside (2011).

Lustig, Roussanov and Verdelhan (2011) again sort portfolios of foreign currencies by

interest rate and identify what they call a “slope” factor, i.e. the difference in average

returns between the highest and lowest portfolios. Using an affine model framework similar

to that of Backus Foresi and Telmer (2001), they show that the slope factor is free from

country specific risk. Rather, it reflects each countries idiosyncratic exposure to a global

shock, and this motivates DRS to derive a “slope-factor” variable of their own using data

on global imbalances. The pricing factors that I derive in this paper are similar.

The literature on risk factors is extensive. Among the relevant papers, Menkoff et al.

(2010) uses a global currency volatility factor as an alternative to Lustig’s et al. (2011).

Also, Rafferty (2010) shows that currencies’ skewness (or crash risk) is an an important

consideration for the investor’s pricing of risk. Burnside, Eichenbaum and Rebelo (2011)

and Burnside (2011) argue that conventional risk factor—however good they might be at

explaining either currency portfolio returns or equity returns—cannot adequately explain

both.

The principal data for this research comes from Lane and Millesi-Ferretti (2004, 2007),

who provide valuable empirical work to document the foreign currency composition of net

assets. They construct a data set with annual estimates of external assets and liabilities

spanning 178 countries for the years 1970-2007. Using this dataset as a starting point, Lane

and Shambaugh (2010) focus on the currency composition of a countries trade volume,

assets, and liabilities over the period 1990 - 2006, and also construct measures for the

foreign currency exposure of the external balance sheet.

2 Gabaix and Maggiori (2013) Model

I start with the nominal model developed in GM. There are two countries: home and for-

eign. Each country has a representative consumer that values tradable and non-tradable

consumption goods. Financiers allow consumers in each country to hedge risk by buying

and selling bonds from each country. Production and money supply are exogenous pro-

cesses, and in the most basic formulation, the model takes place over two time periods. I

will later relax several of these assumption to draw out new results.

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2.1 Households

Household in the U.S. derive utility from consumption across two periods from according

to,

E (θ0ln(C0) + θ1ln(C1)) . (1)

Home consumption in each period Ct is an aggregate of real money balances (Mt/Pt),

home non-traded goods CNt, home consumption of its domestic tradable good CHt, and

home consumption of the foreign tradable good CFt. Mt is the aggregate supply of money

determined endogenously by the central bank, and Pt is the price level that is determined

endogenously. Also, money is non-tradable between countries.

Ct =

[(Mt

Pt

)ω(CNt)

χt(CHt)αt(CFt)

ιt

]1/θt

(2)

For simplicity, θt = ωt + χt + αt + ιt, and each of these terms is non-negative and non-

stochastic. GM do this for tractability and to show with a numerical demonstration that

this assumption is not vital to the results (See Appendix A.3 of their paper). The foreign

country has a similar set up where their two period utility is, E (θ∗0ln(C∗0 ) + θ∗1ln(C∗1 ))

and again, C∗t =[(

MtPt

)ω(C∗Nt)

χ∗t (C∗Ht)ξ∗t (C∗Ft)

ι∗t

]1/θ. Variables with a (∗) above them

correspond to the foreign country.

Consider a full endowment economy so that{YNT,t, YH,t, Y

∗NT,t, YF,t

}is given to the

consumers stochastically and always in positive amounts. Prices in the home market are

{PNt, PHtPFt} and the corresponding prices in the foreign market are {P ∗Nt, P ∗Ht, P ∗Ft}.

With these definitions in hand, in each period the agent will then be confronted with a

static maximization problem,

max (χtln(CNt) + αtln(CHt) + ιtln(CFt)) + λt (Wt − (Mt + PNtCNt + PHtCHt + PFtCFt))

(3)

Here λ is the Lagrange multiplier and Wt is the aggregate consumption expenditure in

time t, which is taken as exogenous. With a linear constraint and concave preferences,

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the first order conditions with equality are,

Mtλt = ωt (4)

PNtCNtλt = χt (5)

PFtCFtλt = ιt (6)

PHtCHtλt = αt (7)

The FOCs for the foreign country are M∗t λ∗t = ω∗t and P ∗NtC

∗Ntλ

∗t = χ∗t and P ∗FtC

∗Ftλ∗ = ι∗t

and P ∗HtC∗Htλ

∗ = ξt. For ease of representation, GM defines Mtωt

= 1λt

= mt. Therefore

home purchases of foreign goods can be written as mtιt = PFtCFt, and foreign purchases

of home goods are mtξt = P ∗HtC∗Ht. Net exports are simply the difference between these

two terms

NXt = etP∗HtC

∗Ht − PFtCFt = etm

∗t ξt −mtιt (8)

Now I can write the Dynamic Maximization problem, where I maximize equation (3)

subject to equation (9) below. As I mentioned above, debt is not traded directly between

the two countries, but rather it is is intermediated through a financier. Debt trading will

be discussed in the next section, but for now interest rates are Rt in the home country and

R∗t in foreign country where, for simplicity, both interest rates are assumed to be strictly

positive and non-stochastic. 3

1∑t=0

1

Rt(YNtPNt + YHtPH)−

1∑t=0

1

Rt(Mt + PNtCNt + PHtCHt + PFtCFt) (9)

Maximizing equation (1) and (3) subject to (2) and (9) yields standard Euler equations

for the choice between consumption in period 0 and period 1. For example, the Euler

equation of our agent for consumption of non-tradable goods is,

1 = E

(βR

UCN1/PNT,0

UCN0/PNT,1

)= E

(βR

χ1/CNtPNtχ0/CNT0PNT,1

)= βRE

(m0

m1

)(10)

3This model can be made richer if instead interest rates are log normally distributed so that R = eµ+σZ andR∗ = eµ

∗+σ∗Z where Z ∼ N(0, 1). Therefore E(R) = eµ+12σ

2

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2.2 Financiers

The other types of agents in this economy are the financiers that facilitates trading of

debt between the two countries. Financiers take long and short positions in home and

foreign nominal bonds, allowing each country to hedge consumption risk but only through

the financier. The financiers balance sheet consists of q0 dollars or q0e0

yen, such that q0

would be the financier’s long position in the dollar, or − q0e0

would be an equivalent short

position in yen. The financier performs this role of intermediary, but imperfectly, as they

are able to divert Γ share of their net balance for their own profits. This imperfection in

the market allows the financiers to make a net gain on their investments that will be a

combination of the interest rate and any exchange rate changes that happen next period.

The financier expects to earn Q0E(R) on dollar denominated debt and Q0q1R∗/q0 on Yen

denominated debt.

Overall the financier is charged with maximizing their value function given by V0,

maxq0

V0 = Q0E

(R− e1R

∗

e0

)(11)

Suppose also that each financier can divert a portion of invested funds equal to Γ Q0

m0q0

and assume a unit mass of financiers, so that investors will be able to recover 1− Γ Q0

m0q0

if the financier reneges on debt obligations. V0 at the least will be Q0 or at most be ΓQ2

0q0

when ΓQ0

q0> 1. Therefore the financier maximizes their profits subject to the constraint,

V0 ≥ min(

1,ΓQ0

m0e0

)|Q0|. (12)

Taking the FOC with respect to Q0 and re-arranging the equations I obtain

Q0Γ = m0E

(e0 − e1

R∗

R

)(13)

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2.3 Market Clearing

The key equations for solving for the equilibrium exchange rate in GM’s extended model

are the equations for bond market clearing at time t = 0 and time t = 1:

NX0 +Q0 −DUSb +DJ

b e0 = 0 (14)

NX1 −Q1R = 0 (15)

where NXt is defined above by equation 8. Intuitively this says that bilateral NFA

positions between two countries must sum to zero. Following the extended GM model,

DUSb and DJ

b are the stocks of debt. These enter into the flow equations at time t = 0

and remain unchanged at time t = 1.

3 Taking the Model Forward

3.1 Debt Stocks

In this section I build on GM and I redefine the existing level of debt stocks denominated

in each currency. In GM, the authors assume that all assets denominated in each countries

domestic currency are also issued by the same county. Here I relax that assumption and

define the existing stock in terms of bonds owned by country i, but denominated in country

j’s currency. Therefore a bond bij has been issued by country i but is denominated in

the currency of country j. To simplify the equations and without much loss in richness,

I assume that m0 = m∗0 = m1 = m∗1. I can re-write the existing stock of debt in terms of

U.S. dollars, I have simply:

Dhb = bhfe0 + bhh (16)

e0Dfb = bfh + e0b

ff (17)

The existence of additional bonds may cause complications with some of the model’s

assumptions. I begin to explore this idea in Appendix A with respect to the UIP condition,

but more work should be done to better understand the implications of these new bonds.

For now I re-arrange the equations above so that I can write debt in terms of U.S. Dollars

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and debt in terms of Yen as follows:

Dus = bhh − bfh (18)

Dj = bff − bhf (19)

The rest of the analysis in GM follows through. In the next section I will extend some

of the GM results in this slightly modified framework, and I show more explicitly the

impact of debt denominated in different currencies. To simplify the presentation for the

equations that follow, I define

ι0 = m0ι0 +DUS (20)

ξ0 = m∗0ξ0 +DJ (21)

ι1 = m1ι1 (22)

Using this straightforward setup, I can plug equations 20, 21, and 22 into GM proposition 3

and their equations 14, 15, and 16 that are reproduced here for convenience, and modified

with ξ1 = 1. Note here that ξ1 is the value of home tradable goods to the foreign consumer

in utility units in period 1, as explained above. See Appendix B for additional details

about these equations and where I obtained them.

e0 =E[ι0 + ι1

R

]+ Γξ0

R∗

E[ξ0 + 1

R

]+ Γξ0

R∗

(23)

E(e1) =R

R∗

E(R∗(ι0 + ι1

R

))+ Γξ0E

(R∗ ι1R

)E(R∗(ξ0 + 1

R∗

))+ Γξ0

(24)

The Returns to the Carry Trade for a long position will be e1R∗

e0R− 1 = Rc. As in GM, I

will look only at the expected returns E(Rc) which is E(e1)R∗

e0R− 1 = Rc. Using the two

equations above, and again setting ξ1 = 1, after some algebra I obtain,

Rc =

R∗

R E (ι1)(

Γξ0 − 1)− Γι0 + ι1

R

ι0 (R∗ + Γ) + R∗

R E (ι1)(25)

This is the key equation of the paper and is similar to Equation 1 of DRS and equation 24

in Proposition 12 of GM. The difference here is that I have only set ξ1 = 1, rather than both

ξ1 and ξ0 as the other authors do. This allows a discussion about net foreign liabilities

11

denominated in domestic currency, rather than simply the share of foreign liabilities in

domestic currency.

∂Rc

∂ι0< 0

∂Rc

bhh< 0

∂Rc

bfh> 0 (26)

∂Rc

∂ξ0

> 0∂Rc

bff> 0

∂Rc

bhf< 0 (27)

Adding and subtracting these terms according to the impact on the Carry Trade will allow

us to say that bff +bfh−bhf −bhh should be strictly increasing along with expected Carry

Returns.

Figures 1 and 2 show the importance of considering the net share versus the simple

share of liabilities denominated in foreign currency. The graphs show each countries

foreign balances sorted by the size of share in total assets and liabilities. The red line

shows the variation in Net balances and how the ordering of countries would be very

different if instead the sample were ordered on net balances. The sorting of countries by

share is the mechanism used by DRS to create Carry Return portfolios and calculating

the IMB factor. It is my goal here to show that the ordering would be quite different if I

loosened the assumptions on the type of debt a country can hold. Relaxing this assumption

is the motivation for my calculation of the IMM and RKP risk factors, described later in

this paper.

3.2 Risk Aversion Approximation

In this section I find a fair approximation of the risk aversion parameter Γ in the GM

model that can be easily calculated from the data. In the GM model, when a country

is a net debtor or has a negative balance of debt denominated in its home currency,

then a negative shock to global risk aversion will depreciate the home exchange rate.

Alternatively, the exchange rate will appreciate under the same shock if the country is a

net creditor or holds a net surplus of assets denominated in the home currency. This is the

hypothesis tested directly in DRS. It assumes that Γ is a parameter subject to exogenous

shocks and exchange rates are endogenous.

I consider the difference between ex-ante and ex-post excess returns. Expectations of

high currency returns will increase the ex-post returns from a stochastic shock to exports,

12

Figure 1

Figure 2

13

imports, preferences or output. Rearranging equation 13, I can write simply that

Γ0 =e0RcQ0

(28)

Using this expression in equation 57, it is straight forward to show that {e1} is decreasing

in Γt. Then I can show that

0 <∂Rc∂Γ0

<∂Rc∂Γ0

when NFA < 0 (29)

0 >∂Rc∂Γt0

>∂Rc∂Γ0

when NFA > 0 (30)

Together with equation 57, these expressions say that debtor (creditor) countries with

higher risk aversion at time t will have higher (lower) ex-post Carry Returns at time t+1,

and countries with lower (higher) risk aversion will have lower (higher) ex-post returns.

The same can be said for expected Carry Trade Returns Rct . In order to apply this

definition of risk aversion to the data, I need to define Q0 in terms of available data. In

the Appendix D, I show that Qt is approximately equal to NFAt in the infinite horizons

version of the GM model proposed in their Appendix 8.

3.3 Aggregate Currency Exposure with Multiple Countries

In this section I decompose the change in the NFA position into two components: exchange

rate changes and the changes in the shares of assets are liabilities. I assume that exchange

rates move independently in the short run from the prices and quantities of the assets

in their home currency. Often exchange rates are modeled as a random walk and shown

empirically to approximate one (e.g. Meese and Rogoff (1983) , Bachetta (2006) and

(2010) among many others), and the well documented failure of PPP is also evidence of

this (Taylor, Peel and Sarno (2001),).

I start by writing the net debt stocks defined in the GM model. Note that in a

two country world, the foreign assets of the home country are the foreign liabilities of the

foreign country. With a financial intermediary, a representative agent in each country, and

with foreign bonds not offered for sale in the domestic market, domestic bonds will not

be sold in the domestic market. That is, all net positions are someone else‘s external net

positions. Given this, NFAt = At−Lt, ωA = A/(A+L), ωL = L/(A+L), ωAab = babebh/A,

ωLab = babebh/L, and variables with hats above them denote percentage changes (X =

14

dX/X). Also note that dNFA/(A + L) = AωA − LωL and∑N

i ωKab = 1. First, I start

with the definition of Net Foreign Assets, NFAt = (bfh + bffefh) − (bhh + bhfefh). In

Appendix E, I show that I can write:

d

(NFA

A+ L

)=

[ωA

N∑i

(ωAhiehi

)− ωL

N∑i

(ωLij eij

)]︸ ︷︷ ︸

%∆ from Exchange Rates

+

[ωA

N∑i

(ωAhibhi

)− ωL

N∑i

(ωLij bij

)]︸ ︷︷ ︸

%∆ from Reallocation

(31)

The first term on the right side of equation 31, lets call this E, shows the percentage

change in the net asset position that is due to exchange rate changes. The second term on

the right side shows the percentage change in the net asset position that is due to changes

in the share of NFA in all destinations, lets call this B. These terms are weighted by the

size relative contribution to changes in the countries NFA. The left hand variable I will

call dNFA.

There are several ways to interpret these terms. At the risk of being too simplified,

E are the portion of prices changes seen by foreign investors only, B is the portion of

price changes that are seen by domestic investors only, and both together are seen by

foreign investors. Also, E is an intuitively clear measure of a countries idiosyncratic

exposure to exchange rate risk, and is almost identical to the Asset Weighted Exchange

rate index proposed and calculated in Lane and Shambaugh (2010). Countries with a

high value for this term will benefit from an exchange rate depreciation and will gain

from an appreciation. Likewise, countries with a low value in this term will benefit from

a currency appreciation, and be hurt by a depreciation. One might also expect that if

short term exchange rates movements are more or less random, then E should lead B in

the data.

I can take this decomposition to the data. d(NFA/(A+L)) and E can be calculated

using the data from Lane and Shambaugh (2010) and the other term, B is whatever is left

over. Figure 3 below shows graphically E, B and dNFA over time for the full sample of

countries averaged for each year. There is a tendency for E to lead dNFA, which can be

seen in Table 1 and also visually in graph 3. There is a clear negative correlation between

B and E that is a product of the simple fact that dNFA is relatively constant while E

moves vigorously.

The tendency for exchange rates and the value of domestic assets to negatively co-

15

Figure 3: NFA Changes decomposed into Price and Quantity Changes

(a) The three terms in Equation 79 are shown parsed out for the years 1992 to 2005 using data fromLane and Shambaugh (2010) .

move is well documented in the literature. The theory, as discussed by Hau and Rey

(2006), is that investors target a level of risk rather than necessarily always higher returns.

Therefore, when exchange rates appreciate, an investor‘s portfolios is more exposed to

exchange rate risk causing the investor to sell assets which lowers the share and price of

the asset. This mechanism is related to the literature on the Equity Carry Trade (see Hau

and Rey (2006, 2008) and Bacchetta and Van-Wincoop (2010)).

Table 1: Correlations for dNFA, E and B

E B dNFA lB lE ldNFA

E 1.00 -0.93 -0.14 -0.65 0.55 -0.51

B -0.93 1.00 0.49 0.54 -0.44 0.44dNFA -0.14 0.49 1.00 -0.11 0.13 -0.00

lB -0.65 0.54 -0.11 1.00 -0.93 0.58

lE 0.55 -0.44 0.13 -0.93 1.00 -0.23ldNFA -0.51 0.44 -0.00 0.58 -0.23 1.00

Lagged variables appear with an ”l” prefix, so that lB, lE and ldNFA are one-year-laggedvariables of B, E and dNFA. The number of observations used in this Table is n = 15, whichmakes it difficult to draw any solid inference. Even so I present to show that the data does notopenly contradict the intuition.

16

4 Data and Currency Portfolios

This section describes the data and empirical procedure used to test the relevance of

equation 25. I start with a detailed description of the data and proceed to describing the

method for calculating each proposed risk factor. The next section tests the risk factors

using a linear factor pricing model.

4.1 Data

Data on Spot and Forward Rates Daily spot and 1-month forward rates are col-

lected relative to the U.S. Dollar from October 16, 1989 to April 23, 2015 via Barclays and

Reuters Datastream 4. There are 55 countries in the full sample pulled from Datastream.

Due to time constraints and possible violations of CIP, I use only a partial sample of 20

countries, which I name Developed-Plus and describe in Appendix G. The subset of 15

developed countries within Developed-Plus are also described in Appendix G.

I also remove large deviations from CIP following the suggestions from Lustig, Rous-

sonav and Verdelhan (2011). These observations are: South Africa from the end of July

1985 to the end of August 1985; Malaysia from the end of August 1998 to the end of June

2005; Indonesia from the end of December 2000 to the end of May 2007; Turkey from the

end of October 2000 to the end of November 2001; United Arab Emirates from the end

of June 2006 to the end of November 2006.

Data on Imbalance Factors The data on external positions is sourced from Lane

and Milesi-Ferretti (2004, 2007) and Lane and Shambaugh (2010). The original data set

is from 1990 - 2004 but is updated to 2012 by Benetrix, Lane and Shambaugh (2014).

While the primary set of data is available for the entire period, the weighted exchange

rate index calculated in Lane and Shambaugh (2010) needed to calculate E is not updated

beyond 2005. All data is reported annually, while exchange rates and forwards data are

reported daily and sampled on the last day of each month. When working with the data

I assume, as does DRS, that all NFA related variables are held constant throughout the

year so that every day of the calendar year has the same observations on imbalances. This

is not unreasonable if the components of the NFA move slowly.

4The University of California, Berkeley was gracious to let me use their facilities

17

Currency Excess Returns Excess returns from going long in the foreign currency

will be

RXt+1 =(St+1 − Ft)

St

=(St+1 − St + St − Ft)

St

=(St+1 − St)

St+

(St − Ft)St

≈ (St+1 − St)St

+ i− i∗

When the CIP condition holds, the forward premium is approximately the interest

rate differential, i− i∗ = (Ft − St) /St. Given that CIP generally holds in the data,5 (e.g.,

Akram, Rime, and Sarno, 2008) the forward discount become fdt = (St−Ft)St

I follow the approach of Bekaert and Hodrick (1993) in calculating excess returns for

each set of portfolios. Observations for forward rates and spot rates on each currency are

sampled at the end of each month (Ft and St), and compared to the spot exchange rate

at the end of the following month (St+1). These three terms are then used to calculate

the excess returns according to equations in 33 below. The m superscript denotes the

midpoint, the a the ask price, and b the bid price. Going long in the foreign currency

means to purchase a forward contract at the ask price at time t for F at , that pays the

foreign currency value in terms of U.S. dollars at time t + 1. Going short in the foreign

currency means to sell a forward contract at F bt that pays the value of the foreign currency

at time t+1. Calculating the excess returns according the following set of rules will account

5For open market macro models, with complete markets it is common to find the standard UIP conditionfor a home and foreign country (denoted by ∗). The condition is:

1 + i

1 + i∗=E(St+1)

St(32)

so that an increase in the exchange rate or a depreciation of the home currency, must compensate foreigninvestors with an increase in the home nominal interest rate i. Taking logs we have approximately i − i∗ =(E(St+1)− St) /St which is the standard UIP condition. The CIP condition replaces E(St+1) with the forwardrate Ft and makes no assumptions about the relationship between realized spot rates St+1 and what the marketpredicts Ft.

18

for transaction costs of buying and selling forwards contracts.

RX lt+1 =

Smt+1 − F atSat

Bought at t and unchanged at t+1 (33)

RX lt+1 =

Sbt+1 − FmtSat

Existing at t and closed at t+1 (34)

RXst+1 =

Sbt+1 − FmtSbt

Bought at t and unchanged at t+1 (35)

RXst+1 =

Smt+1 − F atSbt

Existing at t and closed at t+1 (36)

5 Starting an Empirical Model

Factor CAR: Carry Trade Portfolios I construct five Carry Trade portfolios that

are re-balanced monthly to calculate the CAR returns 6. Currency pairs are sorted into

the portfolios on the basis of their forward discounts, where portfolio 1 is the country

pair with the lowest forward discount, and portfolio 5 contains the country pair with the

highest. Excess returns are then calculated for each currency pair in the portfolios, where

each pair yields equally weighted returns. The Carry Trade Strategy is then to take a

short position in the currencies in portfolio 1, and a long position in the portfolio 5. The

returns to the Carry Trade Strategy, CAR, is calculated as the average excess returns to

going long in portfolio 5, plus the returns from going short in portfolio 1.

Factor MOM: Momentum Portfolios I construct five momentum portfolios to

calculate the risk factor, MOM. Each currency pair is ranked each month according to

the excess returns for that pair over the past 3 months (from t − 1 to t − 3). In the

current month the bottom 20% of pairs have the lowest average excess returns and are

assigned to portfolio 1, while the top 20% of currencies are assigned to portfolio 5 and

likewise for the intermediate portfolios. At the end of each month, new excess returns are

calculated based on the performance of the currencies that month. This method follows

the approach of Asness et. al. (2013) , and is similar to the approach of Jorda and Taylor

(2012) , DRS and Menkhof, Sarno, Schmeling, and Schrimpf (2012) . Currencies that do

not appear in all three of the prior months are omitted from the strategy that month.

6This is equivalent to the High-Low strategy, so to simplify the acronyms I call both “CAR”.

19

Factor VAL: Value Portfolios Portfolios I construct five Value Portfolios

to calculate the risk factor VAL, where portfolios are sorted on the deviation from

the equilibrium nominal spot exchange rate and can be thought of as a measure of

real exchange rate misalignment. Following Jorda and Taylor (2012) and Menkhof,

Sarno, Schmeling, and Schrimpf (2012) , the approach is to sort currencies based on

log(St)/ log(St−1) − log(Pt)/ log(Pt−1) log(P ∗t−1)/ log(P ∗t ), where St is the spot exchange

rate of the home currency to the foreign and Pt is the aggregate price level that is proxied

by a consumer price index. I compare the current month’s data to a moving average of

the spot exchange rate and price level between 4.5 and 5.5 years prior.7

Factor DOL: Dollar Risk Portfolios The Dollar Factor (DOL) is calculated as the

returns on a strategy that goes short in the U.S. Dollar and long in every other currency.

This represents the risk of using the Dollar as a base, and is shown in Lustig, Verdelhan

and Roussonov (2011) to represent the excess returns in payment for country specific risk.

This measure is equivalent to the returns of going short on the dollar and long in all other

currencies.

Factor IMB: Imbalance Portfolios I follow DRS in constructing portfolios that

are first sorted on the Net Foreign Asset position as a percent of nominal GDP of the

investment country (YNFA) and then re-sorted on foreign liabilities denominated in the

domestic currency of the investment country (LLDC) as a share of total liabilities. These

data are available for download from Phillip Lane’s website and results from the data are

published in Benetrix, Lane and Shambaugh (2014) .

Using the YNFA for all countries and years in the data set of Benetrix, Lane and

Shambaugh (2014) , I split the currencies that appear each month into groups above and

below the median 8. Those currencies that fall below the median YNFA, I sort into five

quintiles based on their LLDC. The bottom two quintiles have the lowest LLDC, and are

placed into portfolio 5. The next two lowest quintiles are placed in portfolio 4. Finally

the top quintile is place in portfolio 3. The same thing is done for currencies with an

YNFA that falls above the median. Again these currencies are sorted into quintiles where

the two quintiles with the highest LLDCs are placed into portfolio 1, the next two lowest

7At this stage of research I was only able to obtain estimates of U.S. inflation for all periods at the monthlylevel. Therefore what I use as my measure of VAL is log(St)/ log(St−1)− log(Pt)/ log(Pt−1)

8the median country is included above group

20

are placed into portfolio 2, and finally the lowest quintile is placed in portfolio 3. This is

the same procedure outlined in detail in DRS and in my Appendix C.

An alternative way of sorting is to define the percentiles from the set of currencies

that appear each month in the data set, rather than defining the percentiles based on all

countries over all time periods. Because currencies come into and leave the sample more

often, returns are generally lower due to adjustment costs.

Factor IMM: Adjusted Imbalance Portfolios I perform the same double sorting

procedure described above, but I have changed slightly the definitions of NFA and LDC

so that I can use the same definition of NFA throughout the empirical application. DRS

uses NFA scaled by the GDP in the foreign country, and the LDC is the foreign liabilities

in domestic currency scaled by total liabilities. Here I use NFA and the LDC scaled by

the total assets and liabilities of the foreign country. I will refer to them throughout the

paper as Adjusted NFA and Adjusted LDC, or simply ANFA and ALDC.

ANFAt =

(bhf + bhh − bff − bfh

)(bhf + bhh + bff + bfh)

(37)

ALDCt =(bff − bhf )

(bhf + bhh + bff + bfh)(38)

In stead of the definitions used in DRS,

Y NFAt =

(bhf + bhh − bff − bfh

)Y ft

(39)

LLDCt =bff

(bff + bfh)(40)

Notice the Adjusted LDC is a measure of the net share of liabilities in domestic currency.

This adds an extra penalty (in terms of likely exchange rate depreciation) for any country

with a high level of liabilities in foreign currency.

Factors ANFA and ALDC: Adjusted Definitions The YNFA and LLDC are

tested separately in the DRS paper using their definitions of the equations 39 and 40.

Following their example I also test my definitions of ANFA and ALDC shown in equations

37 and 38. I do this for completeness and comparability to their work.

Factor ERE: Exchange Rate Exposure Following equation 31, a portion of the

NFA-changes can be seen in an exchange rate index weighted by currency composition of

21

foreign asset holdings.

[ωA

N∑i

(ωAhiehi

)− ωL

N∑i

(ωLij eij

)](41)

In Lane and Shambaugh (2010), this measure is a sort of hybrid between their equation

4 and equation 8, showing both aggregate level of exchange rate risk in the Net Foreign

Position, and the change in this overtime.

I construct 5 portfolios based on expression 41, where portfolios with the lowest mea-

sure are the least ”exposed” and therefore placed in portfolio 1. Currencies with the

highest value are placed in portfolio 5. Looking to graph 3, this general intuition seem

to hold true as E slightly leads the NFA. An investor goes long in portfolio 5 currencies

and short in portfolio 1 currencies. ERE returns are calculated as the excess returns on

portfolio 5 minus the excess returns on portfolio 1.

Factor IERE: Percentage Changes in Percentage Changes Taking the re-

ciprocal of the ERE factor, I have the factor IERE. The intuition behind doing this is to

give an extra penalty to countries with large imbalances while also having large balance

sheets. Formally there is some justification as it is an approximation of the first derivitive

of dlog(NFA/(A+L)) shown in equation 31 with respect to the exchange rate. I assume

d2(NFA/(A+ L)) is constant so I am left with the reciprocal of ERE.

Factor UERE: Un-weighted Exchange Rate Exposure In the spirit of testing

alternatives, I also test the unweighted exchange rate exposure. This measure is different

from ERE because it assumes that ωA and ωL are one half. The measure on which I sort

portfolios is then,

[N∑i

(ωAhiehi

)−

N∑i

(ωLij eij

)](42)

Factor RKP: Risk Premium Equation 28 is a rough estimate for the country spe-

cific risk aversion. I construct 5 portfolios with currencies double sorted on the Adjusted

NFA and the risk premium Γ. Above I describe how to calculate Γ from the data us-

ing the expected Carry Trade returns and the NFA from period t. When applying this

definition to the data, I assume that Carry Returns follows an AR(1) process where the

expected value of next periods Rc is proportional to last periods period Rct . For the time

22

being, I assume the proportion is 1 and the constant in the AR(1) regression is zero. In

later research I will estimate the true regression’s parameters, or take the average of the

realized returns from multiple past periods. The Γ used at this stage in research is then:

Γt ≈etR

ct−1

NFAt(43)

Factor DKP: Normal Random Variable As an informal falsification test, I gen-

erate observations following a normal distribution with mean and variance set equal to

the mean and variance of the adjusted NFA across all years and countries. The results

are reported along with the other factors in the Appendices.

6 Empirical Methodology

In the absence of arbitrage opportunities, excess returns must have a price of zero after

taking risk into account and must satisfy the Euler equation:

E (Mt+1RXt+1) = 0 (44)

The SDF, Mt+1, can be approximated by a linear relationship between pricing factors and

factor loadings,

Mt+1 = 1− b′ (ft+1 − µ) (45)

where ft+1 are the pricing factors, and b is a vector of the factor loadings and µ are the

means of the factors. This framework can be put in the standard language of a beta pricing

model following Fama and MacBeth (1993). Using equations 45 in 44, and applying the

definition of covariance and re-arranging terms I obtain

Et−1 [RXtk] =F∑j=1

bjCovt−1 (fjt, RXtk) (46)

= cov(f ′t , RXt

)b = cov

(f ′t , RXt

)var(ft)

−1︸ ︷︷ ︸β

var(ft)b︸ ︷︷ ︸λ

(47)

where β is an n × k matrix of factor betas and λ is a k × 1 vector of risk premia. The

two pass estimation procedure for λ and β involves an OLS time-series regression of each

23

portfolio‘s excess returns on the panel of risk factors to yield a n × k matrix of factor

betas.

RXit = αi + f′tβi + εit t = 1, ..., T for each i = 1, ..., n (48)

The second step is a cross section regression of the average expected returns on the

estimated betas from the first stage

E[RXj

]= β

′jλ+ αj (49)

This regression yields estimated results, λ and ˆalpha = RX − βλ.

The first stage time series regression explains excess returns by each portfolios cor-

relations with the risk factors. This yields a matrix of βs that represent each portfolios

exposure (or “loadings”) on the risk factors. In the second stage, I determine the average

portion of excess returns explained by each factor. This is the premium awarded to the

asset for a unit exposure to each factor, i.e. the λ vector.

This technique for explaining excess returns is popular in the literature. Two alter-

native methods would be (1) stacking the j portfolios and the t periods and running a

pooled OLS regression, or (2) taking a cross sectional regression of the average excess

returns on the average factors over time. In the first case, the standard errors are likely to

be wrong because of time series correlation between the portfolios (due to business cycle

fluctuations). In the second case, the estimates are loosing valuable information about

the variation of factors overtime, even though we are correcting for this cross-correlation.

6.1 Robustness

There are variety of methods to check the robustness of the OLS results, including GMM

and GLS. A primary assumption of the linear factor model is that errors between factors

are uncorrelated, which is unlikely given that I am working with time series data. There-

fore I follow the literature in calculating standard errors corrected for heteroskedacity,

non-normality and the fact that the β matrix in the second stage cross section regression

is estimated.

The model can be tested on several criterion. First, if the model in equation 47 is

true then α should be zero and the risk free rate of return to our investor should be

24

zero (taking into account the SDF). The test statistic for the null hypothesis that all cross

sectional pricing errors are jointly zero, α = 0, is derived in Cochrane (2005) and Burnside

(2011) who both offer a clear discussion on testing the significance of the estimated pricing

errors. I follow closely Appendix 7 in Burnside (2011) to calculate this test statistic. The

reported P-values for the statistic in E show the likelihood of observing these values given

that the model is true. Higher values are better news for the model‘s validity. Second,

the overall measure of fit, R2, can be calculated following Equation 12 of Burnside (2011).

Finally, for the first-pass matrix of β and λ, I report heteroskedastic and auto-correlation

corrected errors as well as Shanken corrected standard errors (Shanken, 1992) .9

The GMM procedure described in detail in Cochrane (2005) and Burnside (2011)

estimates the model using the moment conditions,

E(Rt[1− (ft − µ)′ b

])= 0 (50)

E (ft − µ) = 0 (51)

I use an identity weighting matrix for the first stage of GMM. In the second stage, I

use the optimally chosen weighting matrix based on heteroskadastic and auto-correlation

consistent (HAC) estimates of the long run covariance matrix of the moment conditions.

The parameter values for β, λ and the R2 result from the first stage of GMM should be

identical to the OLS estimates when the weighting matrix is the identity matrix. The

second stage of GMM will yield different coefficient estimates and standard errors. For

this reason, and also for comparability to the procedure used in Burnside (2011) , I report

the results for both stages of GMM in the appendices.

7 Descriptive Statistics on Portfolios

I am testing the pricing properties of 14 portfolios sorted on different measures of global

risk. The factors derived from these portfolios are: CAR, MOM, IMB, IMM, VAL, DOL,

ANFA, ALDC, ERE, IERE, UERE, and IUERE, RKP, and DKP. Below I report basic

descriptive statistics for each set of portfolios. In Appendix F I plot bar graphs showing

the mean annualized excess returns on the portfolios.

The results look as expected for the majority portfolios, with generally increasing

9I intend to also calculate Newey and West standard errors

25

excess returns moving from portfolio 2 to portfolio 5. Note that the DKP portfolio has

currencies sorted on a normal random variable with the mean and variance equal to that

of ANFA for the full sample. The returns in Portfolio 1 are reported as if the investor is

going short, so that a large return is reported for several portfolios (CAR, ERE, UERE,

IUERE). Most other portfolios (IMB, IMM, ERE, IERE, UERE, IUERE, DKP), register

large losses in returns in this portfolio.

Another dimension by which to compare strategies is to compute Sharpe ratios and

optimal weights to minimize the risk across a set of strategies. To solve for the global

minimum volatility portfolio, I must minimize ω′Σω subject to the constraint that the

weights sum to unity, ω′ι = 1. Here ω is the N × 1 matrix of portfolio weights, Σ is

the N × N co-variance matrix of asset returns, and N is the number of portfolios to be

assigned weights. I report the weights and Sharpe ratios for each portfolio below in table

4 and 5. For the sample with 20 countries, the Sharpe ratio on CAR and IMB (0.77 and

0.30) are close to those reported in DRS (0.86 and 0.64). Moving to a larger sample of 55

countries in Table 5, however, does not align the shares with those reported in DRS.

Graph 4 reproduces Figure 7 from DRS but with the set of factor returns discussed in

this paper. I show this set of portfolios, along with the standard portfolios, CAR, VAL

and MOM, plotted in the space of annualized expected returns versus standard deviations.

The DPK strategy does not appear on the graph because expected returns are negative.

The mean variance frontier is also plotted as the solid line left of the points, and represents

the set of optimal means and variances for a portfolio that optimizes the trade-off between

returns and risk.

26

Figure 4: Mean Variance Frontier with Portfolio Returns

(a) The Mean-Variance frontier is here plotted with points for each portfolio plotted in the space ofexpected returns and standard deviations. This plot combines the descriptive statistics in this section.

27

Table 2: Summary Statistics for Portfolios

Portfolios Sorted on Forward Discount, CAR

Statistic P1 P2 P3 P4 P5mean 0.30256 -3.81807 1.85200 3.06410 5.36261

median -0.31009 0.00000 0.00000 0.47203 7.78982standard deviation 8.83956 9.17725 8.78026 9.48267 8.55671

skewness 0.01903 -0.42290 0.22576 0.12098 -0.74839

Portfolios Sorted by Momentum, MOM

mean -2.76279 2.00571 1.85867 1.86389 2.50214median -0.18599 0.00000 0.00000 0.00000 2.02545

standard deviation 9.88096 8.29311 8.70992 8.90622 8.84894skewness 0.09486 0.10340 0.01305 0.12107 0.01151

Portfolios Sorted by Account Imbalances, IMB

mean -2.76279 2.00571 1.85867 1.86389 2.50214median -0.18599 0.00000 0.00000 0.00000 2.02545

standard deviation 9.88096 8.29311 8.70992 8.90622 8.84894skewness 0.09486 0.10340 0.01305 0.12107 0.01151

Portfolios Sorted by Adjusted Imbalances, IMM

mean -1.61966 0.97045 1.32343 1.51844 3.42735median 0.00000 0.00000 1.01096 2.76878 5.08858

standard deviation 9.73101 8.40929 9.70856 10.14993 9.51514skewness -0.03196 0.23186 -0.01563 -0.37686 -0.39108

Portfolios Sorted on Value, VAL

mean -3.51784 -2.71131 -0.31613 2.65132 3.03443median -0.20893 0.00000 0.00000 0.00000 5.00634

standard deviation 9.53859 8.21431 9.60455 8.71137 8.60955skewness -0.11925 -0.90420 -0.07019 -0.17668 -0.70728

Portfolios Sorted on Dollar Risk, DOL

mean 0.00000 0.00000 0.00000 0.00000 1.79106median 0.00000 0.00000 0.00000 0.00000 1.41720

standard deviation 0.00000 0.00000 0.00000 0.00000 8.63210skewness -0.17823

Portfolios Sorted on Risk Premium, RKP

mean -4.34348 -0.80286 1.41249 2.07318 2.05944median -5.52046 0.00000 0.00000 0.00000 0.75647

standard deviation 9.86582 8.50505 9.36321 8.89696 8.84174skewness 0.86597 -0.16868 -0.02692 0.10810 -0.06770

28

Table 3: Summary Statistics for Portfolios

Portfolios Sorted on Adjusted NFA, NFA

Statistic P1 P2 P3 P4 P5mean -4.80400 2.36893 1.82131 1.78558 1.28732

median -6.73753 1.42054 1.02501 1.43020 0.14506standard deviation 9.67866 9.85204 8.86225 9.79270 8.96585

skewness 0.73791 -0.14259 -0.19496 -0.01743 0.26085

Portfolios Sorted on Adjusted LDC, LDC

mean -1.23364 0.86547 1.07560 4.88884 3.71111median 0.00000 0.00000 1.53062 3.69563 4.90585

standard deviation 10.28328 9.20148 9.41756 10.40362 8.42543skewness -0.23601 0.04508 0.09551 -0.37615 -0.76797

Portfolios Sorted on Aggregate Exposure, ERE

mean 5.54354 -1.58634 3.56262 4.81901 5.33647median 3.14948 -1.67469 0.00000 2.33994 2.31162

standard deviation 7.34751 8.58134 7.91936 7.81298 8.90507skewness 0.02151 0.19302 0.61996 0.10115 0.59858

Portfolios Sorted on Inverse Aggregate Exposure, IERE

mean -1.55848 1.56542 -1.18772 0.29499 3.85745median -2.07254 0.00000 -3.14948 0.00000 0.00000

standard deviation 5.73826 9.78084 8.86498 8.84014 7.58804skewness 0.20876 0.33895 0.72811 0.20065 0.50048

Portfolios Sorted on Unweighted Aggregate Exposure, UERE

mean 7.29846 -2.44703 2.45524 4.49623 8.91717median 2.59002 -2.56845 -0.49337 4.22791 6.34343

standard deviation 8.37936 7.52455 8.49795 10.40940 8.40307skewness -0.01015 0.01222 0.48539 1.49167 0.30398

Portfolio Sorted on Inverted UERE, IUERE

mean 2.05893 4.16054 0.66306 3.83307 3.09061median 1.48071 0.94069 0.00000 4.12602 0.00000

standard deviation 7.53206 8.20468 8.98780 10.10650 8.09927skewness -0.43541 0.54561 0.25204 0.14635 0.71985

Portfolio Sorted on a Normal Random Variable, DKP

mean -3.31843 2.68932 1.40815 1.07390 1.30281median -3.90583 1.93947 1.44306 0.07665 1.37727

standard deviation 9.13959 9.53609 9.51841 9.72303 9.10591skewness -0.02535 -0.24377 -0.06363 -0.32972 -0.17214

29

Table 4: Summary Statistics: Portfolio Weights and Sharpe Ratios20 Countries

Portfolio Portfolio Share (ω) Sharpe RatioCAR 0.11 0.77

MOM 0.04 -0.02IMB 0.15 0.30IMM 0.08 0.07VAL -0.01 -0.04RKP 0.05 0.40DKP -0.00 -0.38NFA 0.31 -0.59LDC 0.21 0.16ERE 0.04 1.27

IERE 0.07 0.20UERE -0.03 1.96

IUERE -0.01 0.63

Table 5: Summary Statistics: Portfolio Weights and Sharpe Ratios55 Countries

Portfolio Portfolio Share (ω) Sharpe RatioCAR 0.03 1.21

MOM 0.01 0.05IMB 0.21 0.26IMM 0.04 0.21VAL 0.04 -0.29RKP 0.06 0.06DKP 0.05 -0.66NFA 0.30 -0.51LDC 0.02 -0.22ERE 0.08 0.22

IERE 0.15 0.88UERE 0.01 1.98

IUERE -0.01 1.05

30

8 Results

This paper develops a set of risk factors and trading strategies that are improvements upon

the empirical work of DRS. I report the results for all imbalance strategies developed in

this paper (ERE, IERE, UERE, IUERE, IMB, IMM and RKP) in the data appendix, and

for comparability, I report the results from Burnside (2011) on a three factor model using

quarterly data on consumption growth, durable consumption growth, and average U.S.

equity returns 10. I also report the results from several trading strategies (CAR, VAL and

MOM) to show that my results are broadly comparable to the results of DRS. The results

demonstrate that my risk factors had varying success in explaining returns.

In Appendix E, I report the main results 11, with the estimated bs, λs, pricing errors

α, R2 and HAC standard errors for the factor pricing models. First, I reproduce several

of the results in Burnside (2011) to verify that my methodology is correctly. Second, the

IMB factor performs relatively well, with positive signs on all bs except those calculated

using the ERE portfolios. The R2 and pricing error measures are similar to those reported

by DRS, except for the CAR measure. Unfortunately, standard errors calculated by GMM

are insignificant for all bs except for the IMB factor on the IMM portfolio.

RKP and CAR perform about as well across the same host of measures, only mispricing

the ERE portfolios, with almost significant standard errors for several of the bs. None of

the λs are significant, while the R2 values are higher across all portfolios with the exception

of ERE. The similar performance is promising for the RKP. It is true the ANFA is part

of the definition for RKP, and is used to compute the IMB returns and that the returns

to IMB and RKP are similar.

UERE, IERE, IUERE, and IMM do about equally well (or poorly). The bs are the

wrong sign for three of the factors, and are insignificant except when priced against the

portfolios sorted on ERE. The λs for these factors are insignificant, while the R2 for the

regressions are quite mixed depending on the portfolio. For the best known portfolios, i.e.

VAL, MOM, CAR and IMB, the R2s are generally below 0.7. ERE has a negative signs

for 5 of the 8 portfolios against which it is priced, and the R2 is negative for four of the

portfolios. Overall, ERE does the poorest of the set of factors tried in this paper. Looking

10See my discussion on the empirical procedure for additional details about the data, or refer to Burnside(2011) or Lustig Verdelhan (2007)

11These results mirror those reported in Table 7 in DRS, but with the additional factors described in thispaper

31

above to the descriptive statistics on returns, the Sharpe ratios for ERE and UERE are

unusually, or even impossibly high. The Sharpe ratios for VAL and MOM are below those

generally reported in the literature on this topic.

Overall, the results are varied and not promising for my proposed factors. Generally,

the results indicate that IMB, CAR and RKP are best able to explain the excess returns

from currency trades. However, given odd results for the Sharpe ratios and the inconsis-

tent signs on the pricing betas, I hope for better results with an improved dataset that

exactly matches that used by DRS. I discuss this further in the next section, as I describe

extensions and ways to improve this paper.

9 Extensions

As pricing factors, RKP and CAR perform approximately as well as the IMB factor

proposed by DRS. I should note that the sample size is limited and not directly comparable

to DRS. Their sample extends from 1983 to 2014 and includes 55 countries. In this

paper, I used data from 1990 to 2004 (for ERE, IERE, UERE, IUERE) and to 2012 (for

CAR, VAL, DOL, RKP, DKP, IMB, and IMM). The data set in Lustig Roussanov and

Verdelhan includes only 35 countries but extends for a longer time period. In Menkhoff,

Sarno Schmeling and Schrimpf (2012) 48 countries were used and the time period was

again longer.

Second, I should search the data for severe CIP violations. Looking to Tables 4 and

5, the Sharpe ratios for some portfolios are well above 1, and close to 2 for the UERE

factor. These results are almost surely an error or the result of outliers. Currently, I only

remove the observations that were listed in Lustig, Roussanov and Verdelhan (2011) as

being in severe violation of CIP.

Third, I should include Newey and West (1987) standard error in all regressions for

additional robustness. There are also a number of robustness checks that I should perform

to approach the level of thoroughness of DRS. I should include results for (1) portfolios

sorted on real interest rate differentials (2) removing iliquid currencies according to the

BIS Triennial Survey, (3) alternative base currencies (4) individual currencies, via the

comments by Ang, Liu and Schwarz (2010) 12,(4) I should verify that equation 43 does

12These authors argue that currency portfolios throws away valuable information on variation in the β, andtherefore it is important to test factor models against single assets

32

not impact my results in a meaningful way and finally, (5) I should run an auto-regression

of Rct to improve my RKP factor or at least justify my assumptions.

A

The standard GM model with money and debt stocks only provides two bonds, bhh,

and bff that pay returns next period equal to Rh and Rf . This appendix explains the

theoretical implications for additional bonds, bfh and bhf that will pay returns Rhf and

Rfh. The nature of these bonds are such that,

RffE(e1) = Rfh (52)

RhhE(e1)

= Rhf (53)

This will not be a problem for determining the exchange rate because there are market

imperfections inherent in the model through imperfect risk sharing because Γ > 0. Re-

writing equation 13

Q0Γ = m0E

(e0 − e1

R∗

R

)= m0E

(e0 −

Rfhe1

RhhE(e1)

)

so that after re-arranging the terms,

(e0 −

ΓQ0

m0

)RhhE(e1)

e1= Rfh

Similarly we have for the other bond and return, ΓQ0 = m0E(e0 −

Rff e1RhfE(e1)

)and(

e0 − ΓQ0

m0

)−1 Rff e1E(e1) = Rhf . Taking logs of equation 54 and simplifying we obtain

ln(E(e1))− st+1 = ifh − ihh − ln(e0 − ΓQ0) (54)

The log of the spot exchange rate at time t is st and the of the interest rate Rij is iij .

After slightly re-arranging and employing the definitions of returns, we find the simple

33

relationship,

st+1 − st = ihh − iff (55)

where st = ln (et − ΓQt) 6= st. This simply says that the UIP does not hold in our

extended model.

B

For the convenience of the reader I reproduce the full set of exchange rate equations used

in the GM. The definition next period ex-post exchange rates are

e1 = E (e1) + {e1} (56)

where E (e1) follows the equation 24 defined in the text and {e1} is defined as,

{e1} =

{ι1ξ1

}+R

ι0 − E(ξ0R∗ι1ξ1R

)E(R∗(δ0 + 1

R∗

))+ Γξ0

{ι1ξ1

}(57)

and the operator {} is a short hand way of expressing {X} = E (X)

34

C

As described in the text, this table shows the double sorting procedure used to calculate severalof the factors in this paper. As in DRS, I use five portfolios where the center third portfolio isa mixture of: the middle two deciles of currencies sorted on NFA positions, the bottom quintileof currencies sorted on LDC positions for those countries that fall above the median NFA, andthe top quintile of the currencies sorted on their LDC positions for countries below the medianNFA

35

D

First using Appendix 1 of GM, we start with their bond market clearing equation for an

infinite horizon’s version of the same model so far investigated in this model.

NXt −RQt−1 +Qt = 0 (58)

for any period t. Iterating forward, setting Q0 = 0, and recalling that NXt = ιt − exit,

we find that

Qt =

t∑k=0

Rk(NXt−k

)(59)

(60)

It would be useful to put this expression in terms of observable data in order to further

test the intuitions so far discussed. NXt can be re-defined in this new framework, using

the definitions of ιt and ξt given in equations 20, 22, and 21. In the equation below, ft

and f∗t are flows of bonds per period towards consumers in the home and foreign country,

respectively. That is, ft is the net flow of bonds originating from the U.S. to Japan (bhht

and bhft ) in time t, while f∗t is the net flow of bonds originating from Japan (bfft and bfht )

and flowing to the U.S. in time t.

NXt = ιt − exit (61)

= etm∗t ξt −mtιt︸ ︷︷ ︸

Net Exports

+ f∗ − etft︸ ︷︷ ︸NetIncome

+ (R− 1)(DUSt − etDJ

t−1

)︸ ︷︷ ︸Valuation Effect

(62)

= NXt +NIt + V At (63)

= CAt + V At (64)

Where NXt, NIt and V At represent their counterparts in the National Accounts data

for Net Exports, Net Income,and Valuation Effects (arising from interest rate income

and exchange rate re/devaluations. Where does this equation leave us? Starting with

equation 1 of Gourinchas and Rey (2007), the Net Foreign Asset position can be defined

as a function of last periods Net Foreign Asset position, the net export balance, and

interest payments on the net position. Starting with the first equation below, Gourinchas

36

and Rey (2007) add and substract NAt and NIt,

NAt+1 = RtNAt +NXt (65)

∆NAt+1,t = (R− 1)NAt +NXt +NIt −NIt (66)

= CAt + V At (67)

= NXt (68)

Where the second to last line follows because CAt = NXt +NIt, and the last line follows

using equation 67 to see that NXt = ∆NFAt+1,t.

Iterating equation 65 we have

NAk =

t∑k=0

Rk(NXt−k

)(69)

Because I have shown that NXt = ∆NAt, I have from 69 and 59 that Qk = NAk

So if we wanted to measure the time varying exposure of an investor for any currency

pair, then we would rearrange equation 13, and assume for the moment that et+1 is far

enough out in the future so that investors have accurate expectations, and we make the

substitution using Qk = NAk, then we have measure for country specific risk aversion

that is easily observable from the data.

Γt =et − et+1

(R∗

R

)NAt

= − etRc

NFAt(70)

E

I start by writing the net debt stocks. Note that in a two country world, the Foreign

Assets of the U.S. are the Foreign Liabilities of Japan. With a financial intermediary, a

representative agent in each country, and with foreign bonds not offered for sale in the

domestic market, domestic bonds will not be sold in the domestic market. That is, all

net positions are external net positions.13 Given this, NFAt = At−Lt, ωA = A/(A+L),

ωL = L/(A + L), ωAab = babebh/A, ωLab = babebh/L, and variables with hats above them

denote percentage changes (X = dX/X). Also note that dNFA/(A + L) = AωA − LωL

and∑N

i ωKab = 1

13Proove this

37

NFAt = (bfh + bffefh)− (bhh + bhfefh) (71)

=N∑i

bhieih −N∑j

bijejh (72)

=A

A

N∑i

bhieih −L

L

N∑j

bijejh (73)

= AN∑i

ωAhi − LN∑j

ωLij (74)

d

(NFA

A+ L

)= ωA

N∑i

dωAhi − ωLN∑j

dωLij + AωA

N∑i

ωAhi − LωLN∑i

ωLhi (75)

= AωA − LωL + ωA

N∑i

dωAhi − ωLN∑j

dωLij (76)

= AωA − LωL + ωA

N∑i

(ωAhi

(bhi + ehi

)− ωAijA

)− ωL

N∑i

(ωLij

(bij + eij

)− ωLijL

)(77)

= ωA

N∑i

(ωAhi

(bhi + ehi

))− ωL

N∑i

(ωLij

(bij + eij

))(78)

=

[ωA

N∑i

(ωAhiehi

)− ωL

N∑i

(ωLij eij

)]︸ ︷︷ ︸

%∆ from Exchange Rates

+

[ωA

N∑i

(ωAhibhi

)− ωL

N∑i

(ωLij bij

)]︸ ︷︷ ︸

%∆ from Reallocation

(79)

38

F

39

G

Table 6: Countries Included In Samples

Country Datastream Developed-Plus DevelopedArgentina 1 0 0Australia 1 1 1

Austria 1 0 0Belgium 1 1 1Canada 1 1 1

Chile 1 0 0China, P.R. 1 0 0

Colombia 1 0 0Croatia 1 0 0

Czech Republic 1 1 0Denmark 1 1 1

Egypt 1 0 0Estonia 1 0 0Finland 1 0 0France 1 1 1

Germany 1 1 1Greece 1 0 0

Hong Kong SAR of China 1 0 0Hungary 1 0 0

Iceland 1 0 0India 1 0 0

Indonesia 1 0 0Ireland 1 0 0

Israel 1 1 0Italy 1 1 1

Japan 1 1 1Jordan 1 0 0

Kazakhstan 1 0 0Kenya 1 0 0

Korea, Republic of 1 0 0Mexico 1 1 0

Morocco 1 1 0Netherlands 1 1 1

New Zealand 1 1 1Norway 1 1 1Poland 1 0 0

Portugal 1 0 0Slovak Republic 1 0 0

Slovenia 1 0 0South Africa 1 0 0

Spain 1 0 0Sweden 1 1 1

Switzerland 1 1 1Thailand 1 0 0

Tunisia 1 0 0Ukraine 1 0 0

United Kingdom 1 1 1China 1 1 040

H

Portfolio Observations bDOL bERE lambda DOL lambda ERE R2 Alpha Pvalue

ERE 107 0.9413 1.4756 0.0007 0.0013 -0.2016 0.00250.1949 0.3369 0.3497 0.4541

IMB 191 0.2137 5.4168 0.0012 0.0040 0.4035 0.25080.0370 0.4695 0.5507 0.5931

RKP 191 1.4328 1.2429 0.0009 0.0012 0.2277 0.37430.2638 0.1593 0.4340 0.2473

VAL 191 -2.9485 9.0249 0.0005 0.0059 0.1162 0.562-0.4578 0.8699 0.2209 1.0464

MOM 191 -2.3409 10.4985 0.0010 0.0071 0.5618 0.5105-0.2752 0.7433 0.4780 0.9988

CAR 191 -10.1947 23.0521 -0.0000 0.0146 0.0533 0.0102-1.5269 1.9287 -0.0085 3.1933

DKP 191 7.4393 -13.7595 0.0006 -0.0085 -0.0097 0.00771.2194 -1.5856 0.3025 -1.4189

IMM 191 2.7662 -3.2328 0.0006 -0.0018 0.0375 0.18480.5114 -0.5110 0.2930 -0.4105

Portfolio Observations bDOL bIMB lambda DOL lambda IMB R2 Alpha Pvalue

ERE 107 2.1618 -1.4620 0.0008 -0.0006 -0.1492 0.00190.4545 -0.2559 0.3790 -0.1831

IMB 191 2.1709 0.2291 0.0030 0.0100 0.9851 0.26170.6383 0.3448 1.2675 1.1220

RKP 191 2.8127 0.0209 0.0031 0.0088 0.9515 0.17280.8230 0.0315 1.3030 0.9711

VAL 191 1.6488 0.4669 0.0032 0.0121 0.9808 0.7470.4574 0.6239 1.3300 1.3147

MOM 191 4.8884 -0.6139 0.0034 0.0054 0.8755 0.37181.3181 -0.7510 1.4230 0.5670

CAR 191 3.0675 -0.1829 0.0028 0.0065 -0.6704 00.8657 -0.2493 1.1491 0.6948

DKP 191 2.4924 0.0771 0.0029 0.0087 0.8813 0.00080.7251 0.1154 1.2293 0.9547

IMM 191 3.3612 -0.1605 0.0031 0.0077 0.9458 0.20410.9807 -0.2365 1.3181 0.8565

41

Portfolio Observations bDOL bRKP lambda DOL lambda RKP R2 Alpha Pvalue

ERE 107 1.7869 -0.7298 0.0008 -0.0005 -0.1508 0.0020.3977 -0.1564 0.3766 -0.1336

IMB 191 2.8815 3.7336 0.0033 0.0019 0.9527 0.25591.9968 0.3259 1.2876 0.2638

RKP 191 2.8025 2.4943 0.0031 0.0013 0.9519 0.1893.3843 0.6515 1.1620 0.8013

VAL 191 3.4702 2.5881 0.0038 0.0014 0.9386 0.79684.1084 0.3132 1.4463 0.3395

MOM 191 1.9563 17.9211 0.0028 0.0088 0.9411 0.77081.7219 0.8691 1.1089 0.9310

CAR 191 1.0279 25.5872 0.0021 0.0125 0.5482 0.0260.5540 1.8232 0.8155 3.5923

DKP 191 3.5178 -25.6728 0.0028 -0.0123 0.9158 0.39474.5980 -1.3961 1.0718 -1.3968

IMM 191 2.6907 0.3376 0.0029 0.0003 0.8088 0.43561.8157 0.0296 1.1689 0.0365

Portfolio Observations bDOL bUERE lambda DOL lambda UERE R2 Alpha Pvalue

ERE 107 2.3255 1.8788 0.0008 0.0013 -0.4212 0.00410.4843 0.3342 0.3841 0.2738

IMB 191 1.7858 -1.3502 0.0010 -0.0014 0.0988 0.25550.3687 -0.1650 0.4659 -0.2077

RKP 191 2.1650 0.5868 0.0009 0.0002 0.2389 0.45010.4323 0.0734 0.4347 0.0317

VAL 191 0.5362 -2.6124 0.0006 -0.0023 -0.1058 0.33970.1027 -0.4200 0.2888 -0.4598

MOM 191 -0.5065 -9.0015 0.0010 -0.0077 0.5979 0.5786-0.0777 -0.8248 0.4717 -1.0264

CAR 191 -3.0229 -11.5745 0.0002 -0.0096 -0.4105 0.0003-0.5181 -1.4840 0.1155 -1.7959

DKP 191 4.6068 10.5277 0.0006 0.0085 -0.9234 0.00730.8861 0.9704 0.3018 1.0525

IMM 191 2.2866 3.1646 0.0006 0.0024 0.0547 0.17760.4688 0.5379 0.2872 0.4837

42

Portfolio Observations bDOL bDKP lambda DOL lambda DKP R2 Alpha Pvalue

ERE 107 -1.6901 -14.2012 -1.7733 -0.0024 0.5982 0.8332-1.4011 -0.2894 -3.2281 -0.2425

IMB 191 1.0835 38.7328 1.1989 0.0098 0.6112 0.70291.7088 0.3306 1.2067 0.4963

RKP 191 1.0476 56.5766 1.1318 0.0150 0.7947 0.69111.8579 0.8462 1.1957 1.6890

VAL 191 0.9889 26.8896 1.1063 0.0064 0.6898 0.60632.5975 0.4639 1.3741 0.6180

MOM 191 1.5844 63.6092 1.7431 0.0163 0.0425 0.80090.8787 0.6248 1.1729 1.4023

CAR 191 0.5373 93.3037 0.4885 0.0265 -5.5146 0.20660.3015 0.5141 0.3579 1.3293

DKP 191 0.8002 -0.9225 0.9277 -0.0014 0.6768 0.06815.2011 -0.1558 1.0039 -1.0223

IMM 191 0.8580 -23.2788 1.0265 -0.0080 0.6299 0.80292.2615 -0.7426 0.9677 -1.1929

Portfolio Observations bDOL bCAR lambda DOL lambda CAR R2 Alpha Pvalue

ERE 107 1.2005 1.3618 0.0007 0.0013 -0.2481 0.00220.2477 0.2175 0.3604 0.2769

IMB 191 2.9823 7.3488 0.0036 0.0047 0.9406 0.31022.1474 0.2118 1.0215 0.1774

RKP 191 2.8759 2.4094 0.0032 0.0016 0.939 0.19413.4839 0.3021 1.2637 0.2890

VAL 191 3.5298 -0.1861 0.0038 0.0001 0.9192 0.71563.5180 -0.0381 1.5144 0.0186

MOM 191 2.6970 7.2559 0.0033 0.0046 0.8873 0.65651.9886 0.8582 1.3529 0.7746

CAR 191 2.3447 11.7389 0.0031 0.0074 0.3025 0.14591.4507 2.1813 1.2444 3.2113

DKP 191 2.7511 -5.9110 0.0027 -0.0035 0.6858 0.02453.1252 -0.6259 1.1000 -0.4387

IMM 191 2.3483 15.1500 0.0033 0.0095 0.9789 0.82441.4828 1.3451 1.2306 1.7919

43

Portfolio Observations bDOL bIERE lambda DOL lambda IERE R2 Alpha Pvalue

ERE 107 -6.1814 40.3216 0.0007 0.0062 0.574 0.3348-1.3083 2.9131 0.3433 2.9634

IMB 191 5.7207 -17.6822 0.0010 -0.0024 0.4843 0.33210.7707 -0.6936 0.5017 -0.6766

RKP 191 6.4467 -25.0364 0.0007 -0.0036 0.6714 0.79280.7659 -0.7701 0.3583 -0.9399

VAL 191 -0.8184 15.4567 0.0010 0.0025 0.0446 0.4639-0.1602 0.8847 0.4663 0.8843

MOM 191 5.3272 -14.0223 0.0012 -0.0019 0.3742 0.35650.6847 -0.4149 0.5536 -0.3958

CAR 191 -8.8159 68.0625 0.0019 0.0106 -0.0453 0.0457-1.5118 2.6969 0.9197 3.2825

DKP 191 -13.9364 78.3187 0.0005 0.0118 0.1405 0.2278-1.5165 1.7016 0.2433 2.2329

IMM 191 -5.7322 34.1777 0.0004 0.0052 0.4353 0.5199-0.9456 1.2706 0.1823 1.2191

Portfolio Observations bDOL bIMM lambda DOL lambda IMM R2 Alpha Pvalue

ERE 107 1.6500 0.0624 0.0008 0.0004 -0.1386 0.00180.2994 0.0102 0.3699 0.1380

IMB 191 2.1021 0.2442 0.0030 0.0103 0.9864 0.27420.5919 0.3504 1.2649 1.1396

RKP 191 2.8148 0.0201 0.0031 0.0091 0.9514 0.17280.7977 0.0295 1.3038 0.9986

VAL 191 1.5863 0.4785 0.0032 0.0123 0.9806 0.74340.4267 0.6201 1.3350 1.3305

MOM 191 4.9758 -0.6305 0.0034 0.0058 0.8879 0.3781.3147 -0.7602 1.4173 0.6144

CAR 191 3.1416 -0.2007 0.0028 0.0067 -0.6696 00.8636 -0.2668 1.1523 0.7187

DKP 191 2.4982 0.0748 0.0029 0.0089 0.8766 0.00080.7027 0.1084 1.2320 0.9781

IMM 191 3.4282 -0.1751 0.0032 0.0080 0.9543 0.20860.9673 -0.2500 1.3233 0.8872

44

Portfolio Observations bDOL bIUERE lambda DOL lambda IUERE R2 Alpha Pvalue

ERE 107 -1.6331 20.6286 0.0006 0.0054 0.4057 0.0327-0.3464 2.0662 0.3019 2.0512

IMB 191 8.3596 -45.3049 0.0008 -0.0114 0.6097 0.59870.9171 -1.0380 0.3760 -1.3728

RKP 191 10.3109 -56.8536 0.0009 -0.0144 0.9575 0.99581.0129 -0.9356 0.4431 -1.5578

VAL 191 0.3191 9.1478 0.0007 0.0024 -0.0881 0.36240.0462 0.2629 0.3549 0.2750

MOM 191 0.9807 12.9076 0.0013 0.0035 0.4419 0.36150.1703 0.5553 0.5991 0.6045

CAR 191 -13.8984 109.7024 0.0009 0.0281 0.0057 0.5832-1.2242 1.5892 0.4500 3.8398

DKP 191 3.9825 -20.1848 0.0005 -0.0051 -0.3711 0.00830.6677 -0.9346 0.2281 -1.0572

IMM 191 -2.8151 26.4577 0.0005 0.0068 0.1216 0.2532-0.4697 0.9836 0.2280 1.0800

45

I

Table 7: OLS Results for Two Factor Model: dC, dD, rm

OLS Lambdas

lambda se tsts

dC 0.593782 0.733735 0.809259dD 1.095741 1.019365 1.074925rm 0.117447 0.073981 1.587524

OLS Pricing Errors

Statistic

Pricing Error -1.689545Pricing Error -1.011378Pricing Error -1.649906Pricing Error 0.616652Pricing Error -0.899917Pricing Error -1.218173Pricing Error 1.934097Pricing Error 0.340490Pvalue for Pricing Error 0.000798R2 0.339274

OLS Betas

Constant dC dD rm

beta -2.269796 0.200838 0.027553 -6.774078se 1.837996 0.852291 0.611692 5.522009tsts 1.773881 0.636767 0.562573 4.850533beta -2.094887 0.740083 0.091426 -3.416955se 1.918008 0.889393 0.638320 5.762394tsts 1.298244 0.614163 0.466082 6.185582beta -3.122775 -0.639289 0.961859 1.944828se 1.903001 0.882435 0.633326 5.717310tsts 1.830119 1.002104 0.682379 4.729219beta -1.509223 -0.546316 0.982339 -8.852001se 2.362142 1.095341 0.786130 7.096736tsts 2.311995 1.074083 0.759575 6.803255beta -2.128268 0.179653 0.485163 0.938784se 2.168933 1.005749 0.721829 6.516264tsts 2.459080 0.747404 0.708395 6.468895beta -2.828497 -0.754561 1.079084 2.308820se 2.347497 1.088550 0.781256 7.052736tsts 2.471472 0.952061 0.829727 6.769226beta -1.028332 0.035881 1.233557 -2.718612se 2.250623 1.043629 0.749016 6.761690tsts 2.171992 0.812359 0.754948 6.217829beta -1.224341 -1.342072 1.425985 7.876514se 3.609747 1.673864 1.201338 10.844995tsts 1.974705 1.630300 0.761686 11.641580

46

Table 8: Shanken OLS Results for Two Factor Model: dC, dD, rm

Shanken OLS Pricing Errors

Statistic

Pricing Error p -1.689545Pricing Error p -1.011378Pricing Error p -1.649906Pricing Error p 0.616652Pricing Error p -0.899917Pricing Error p -1.218173Pricing Error p 1.934097Pricing Error p 0.340490Pvalue 0.058606R2 0.339274

Shanken OLS Lambdas

lambda se tsts

dC 0.593782 1.010726 0.587481dD 1.095741 1.403392 0.780781rm 0.117447 0.100998 1.162857

47

Table 9: GMM Results for Two Factor Model: dC, dD, rm

GMM b Vector

dC dD rm

beta -0.220391 0.454507 5.164265se 0.624292 0.498163 2.729443tsts -0.353026 0.912366 1.892058beta 0.369970 0.746608 4.650369se 0.448140 0.319834 2.562815tsts 0.825568 2.334364 1.814555

GMM Lambdas

dC dD rm

lambda 0.593782 1.095741 0.117447se 1.030068 1.576868 0.077085tsts 0.576449 0.694884 1.523589lambda 2.372363 3.475942 0.102037se 0.986513 1.102443 0.075183tsts 2.404797 3.152944 1.357190

GMM: Other Statistics

Iteration 1 Iteration 2

Jstat 10.274461 10.274461Degrees of Freedom 5.000000 5.000000Pvalue 0.067822 0.067822R-squared 0.339274 -0.662110

48

Table 10: OLS Results for Two Factor Model: DOL, ERE

OLS Lambdas

lambda se tsts

DOL 0.000719 0.002056 0.349668ERE 0.001255 0.002763 0.454096

OLS Pricing Errors

Statistic

Pricing Error 0.001085Pricing Error -0.002628Pricing Error 0.002708Pricing Error 0.001678Pricing Error -0.000620Pvalue for Pricing Error 0.000646R2 -0.083743

OLS Betas

Constant DOL ERE

beta 0.000667 -1.026705 0.307750se 0.000831 0.041612 0.032879tsts 0.000862 0.050481 0.044319beta -0.002254 0.956450 -0.254533se 0.001033 0.051744 0.040885tsts 0.001059 0.065797 0.051378beta 0.003060 0.989623 -0.186850se 0.000904 0.045256 0.035759tsts 0.000906 0.051046 0.044570beta 0.001950 0.943853 -0.040277se 0.001066 0.053418 0.042208tsts 0.001107 0.055926 0.074560beta -0.000667 1.026705 0.692250se 0.000831 0.041612 0.032879tsts 0.000862 0.050481 0.044319

49

Table 11: Shanken OLS Results for Two Factor Model: DOL, ERE

Shanken OLS Pricing Errors

Statistic

Pricing Error p 0.001085Pricing Error p -0.002628Pricing Error p 0.002708Pricing Error p 0.001678Pricing Error p -0.000620Pvalue 0.000659R2 -0.083743

Shanken OLS Lambdas

lambda se tsts

DOL 0.000719 0.002056 0.349667ERE 0.001255 0.002764 0.454030

Table 12: GMM Results for Two Factor Model: DOL, ERE

GMM b Vector

DOL ERE

beta 0.941254 1.475554se 4.830205 4.379290tsts 0.194868 0.336939beta 1.247899 3.115781se 4.784304 4.021418tsts 0.260832 0.774796

GMM Lambdas

DOL ERE

lambda 0.000719 0.001255se 0.002055 0.002790tsts 0.349824 0.449828lambda 0.001184 0.002503se 0.002070 0.002486tsts 0.571824 1.006526

GMM: Other Statistics

Iteration 1 Iteration 2

Jstat 14.309212 14.309212Degrees of Freedom 3.000000 3.000000Pvalue 0.002513 0.002513R-squared -0.083743 -0.201650

50

Table 13: OLS Results for Two Factor Model: DOL, IMB

OLS Lambdas

lambda se tsts

DOL 0.003035 0.002395 1.267498IMB 0.010036 0.008944 1.122049

OLS Pricing Errors

Statistic

Pricing Error -0.000747Pricing Error -0.000510Pricing Error -0.000441Pricing Error 0.000199Pricing Error 0.000011Pvalue for Pricing Error 0.245444R2 0.989275

OLS Betas

Constant DOL IMB

beta -0.000802 -0.958498 0.195436se 0.000536 0.024305 0.006471tsts 0.000530 0.034003 0.008809beta -0.000453 0.852689 -0.162616se 0.000542 0.024546 0.006535tsts 0.000540 0.026568 0.005087beta -0.000365 1.091393 -0.203499se 0.000583 0.026428 0.007036tsts 0.000583 0.034518 0.006238beta 0.000266 0.954780 -0.176988se 0.000791 0.035852 0.009545tsts 0.000801 0.030076 0.005487beta 0.000802 0.958498 0.804564se 0.000536 0.024305 0.006471tsts 0.000530 0.034003 0.008809

51

Table 14: Shanken OLS Results for Two Factor Model: DOL, IMB

Shanken OLS Pricing Errors

Statistic

Pricing Error p -0.000747Pricing Error p -0.000510Pricing Error p -0.000441Pricing Error p 0.000199Pricing Error p 0.000011Pvalue 0.249199R2 0.989275

Shanken OLS Lambdas

lambda se tsts

DOL 0.003035 0.002395 1.267419IMB 0.010036 0.008944 1.122034

Table 15: GMM Results for Two Factor Model: DOL, IMB

GMM b Vector

DOL IMB

beta 2.170907 0.229112se 3.401141 0.664426tsts 0.638288 0.344827beta 2.207366 0.263093se 3.387025 0.661457tsts 0.651712 0.397747

GMM Lambdas

DOL IMB

lambda 0.003035 0.010036se 0.002378 0.008896tsts 1.276450 1.128149lambda 0.003177 0.010663se 0.002455 0.009477tsts 1.294177 1.125161

GMM: Other Statistics

Iteration 1 Iteration 2

Jstat 3.997490 3.997490Degrees of Freedom 3.000000 3.000000Pvalue 0.261735 0.261735R-squared 0.989275 0.985068

52

Table 16: OLS Results for Two Factor Model: DOL, UERE

OLS Lambdas

lambda se tsts

DOL 0.000758 0.002057 0.368487UERE 0.012522 0.003004 4.168468

OLS Pricing Errors

Statistic

Pricing Error -0.000872Pricing Error -0.003886Pricing Error -0.003181Pricing Error 0.000843Pricing Error 0.005161Pvalue for Pricing Error 0.000001R2 0.413312

OLS Betas

Constant DOL UERE

beta -0.004195 -1.006115 0.702679se 0.000954 0.039641 0.028647tsts 0.000852 0.042945 0.033416beta -0.004099 0.869294 0.111903se 0.001097 0.045548 0.032915tsts 0.001196 0.045576 0.043500beta -0.003685 1.090461 0.195807se 0.001200 0.049842 0.036019tsts 0.001313 0.053595 0.050184beta 0.000435 0.921040 0.161150se 0.001151 0.047788 0.034534tsts 0.001250 0.051748 0.042998beta 0.004195 1.006115 0.297321se 0.000954 0.039641 0.028647tsts 0.000852 0.042945 0.033416

53

Table 17: Shanken OLS Results for Two Factor Model: DOL, UERE

Shanken OLS Pricing Errors

Statistic

Pricing Error p -0.000872Pricing Error p -0.003886Pricing Error p -0.003181Pricing Error p 0.000843Pricing Error p 0.005161Pvalue 0.000009R2 0.413312

Shanken OLS Lambdas

lambda se tsts

DOL 0.000758 0.002058 0.368394UERE 0.012522 0.003035 4.125868

Table 18: GMM Results for Two Factor Model: DOL, UERE

GMM b Vector

DOL UERE

beta 6.440862 15.510585se 5.011603 5.038945tsts 1.285190 3.078141beta 5.410968 24.145423se 4.964597 4.129600tsts 1.089911 5.846916

GMM Lambdas

DOL UERE

lambda 0.000758 0.012522se 0.002070 0.002971tsts 0.366201 4.215444lambda -0.000905 0.020133se 0.002451 0.003784tsts -0.369065 5.320812

GMM: Other Statistics

Iteration 1 Iteration 2

Jstat 19.377458 19.377458Degrees of Freedom 3.000000 3.000000Pvalue 0.000228 0.000228R-squared 0.413312 -0.138935

54

Table 19: OLS Results for Two Factor Model: DOL, MOM

OLS Lambdas

lambda se tsts

DOL 0.003117 0.002458 1.268008MOM 0.000352 0.001978 0.177922

OLS Pricing Errors

Statistic

Pricing Error -0.001522Pricing Error -0.001004Pricing Error 0.000200Pricing Error -0.000879Pricing Error 0.001586Pvalue for Pricing Error 0.345702R2 0.791598

OLS Betas

Constant DOL MOM

beta -0.001704 -0.513035 0.533010se 0.001495 0.045361 0.054558tsts 0.001213 0.270566 0.098705beta -0.000929 0.261931 0.007735se 0.001091 0.033094 0.039805tsts 0.001025 0.122529 0.046867beta 0.000311 0.380684 -0.013060se 0.001372 0.041629 0.050070tsts 0.001209 0.199986 0.048146beta -0.000411 1.600729 -0.092321se 0.002334 0.070818 0.085178tsts 0.001803 0.444256 0.113463beta 0.001704 0.513035 0.466990se 0.001495 0.045361 0.054558tsts 0.001213 0.270566 0.098705

55

Table 20: Shanken OLS Results for Two Factor Model: DOL, MOM

Shanken OLS Pricing Errors

Statistic

Pricing Error p -0.001522Pricing Error p -0.001004Pricing Error p 0.000200Pricing Error p -0.000879Pricing Error p 0.001586Pvalue 0.349943R2 0.791598

Shanken OLS Lambdas

lambda se tsts

DOL 0.003117 0.002459 1.267629MOM 0.000352 0.001978 0.177922

Table 21: GMM Results for Two Factor Model: DOL, MOM

GMM b Vector

DOL MOM

beta 2.906850 0.612418se 1.361667 2.754816tsts 2.134773 0.222308beta 2.257685 0.807636se 0.866319 2.751298tsts 2.606066 0.293547

GMM Lambdas

DOL MOM

lambda 0.003117 0.000352se 0.002216 0.001976tsts 1.406435 0.178064lambda 0.002409 0.000521se 0.001671 0.001979tsts 1.441253 0.263313

GMM: Other Statistics

Iteration 1 Iteration 2

Jstat 3.321250 3.321250Degrees of Freedom 3.000000 3.000000Pvalue 0.344696 0.344696R-squared 0.791598 0.737872

56

Table 22: OLS Results for Two Factor Model: DOL, CAR

OLS Lambdas

lambda se tsts

DOL 0.003133 0.002518 1.244406CAR 0.007364 0.002293 3.211253

OLS Pricing Errors

Statistic

Pricing Error 0.001939Pricing Error 0.000626Pricing Error -0.001661Pricing Error 0.001216Pricing Error -0.000613Pvalue for Pricing Error 0.079408R2 0.646359

OLS Betas

Constant DOL CAR

beta 0.001414 -0.433904 0.296519se 0.001238 0.035567 0.047053tsts 0.001031 0.211448 0.073872beta 0.001594 1.984806 -0.273569se 0.003195 0.091792 0.121435tsts 0.002550 0.585038 0.137984beta -0.001509 0.403296 -0.021827se 0.001487 0.042708 0.056500tsts 0.001426 0.209259 0.070225beta 0.001360 0.439762 -0.006977se 0.001433 0.041164 0.054457tsts 0.001214 0.237739 0.065956beta -0.001414 0.433904 0.703481se 0.001238 0.035567 0.047053tsts 0.001031 0.211448 0.073872

57

Table 23: Shanken OLS Results for Two Factor Model: DOL, CAR

Shanken OLS Pricing Errors

Statistic

Pricing Error p 0.001939Pricing Error p 0.000626Pricing Error p -0.001661Pricing Error p 0.001216Pricing Error p -0.000613Pvalue 0.102521R2 0.646359

Shanken OLS Lambdas

lambda se tsts

DOL 0.003133 0.002531 1.238141CAR 0.007364 0.002334 3.154713

Table 24: GMM Results for Two Factor Model: DOL, CAR

GMM b Vector

DOL CAR

beta 2.344660 11.738887se 1.616261 5.381542tsts 1.450669 2.181324beta 2.866697 17.113332se 1.511720 4.345341tsts 1.896315 3.938317

GMM Lambdas

DOL CAR

lambda 0.003133 0.007364se 0.002307 0.002229tsts 1.357965 3.303959lambda 0.003972 0.010707se 0.002521 0.001900tsts 1.575798 5.636697

GMM: Other Statistics

Iteration 1 Iteration 2

Jstat 5.381523 5.381523Degrees of Freedom 3.000000 3.000000Pvalue 0.145899 0.145899R-squared 0.646359 0.302529

58

Table 25: OLS Results for Two Factor Model: DOL, IERE

OLS Lambdas

lambda se tsts

DOL 0.000594 0.002073 0.286494IERE 0.001915 0.001847 1.036612

OLS Pricing Errors

Statistic

Pricing Error 0.000156Pricing Error 0.001258Pricing Error -0.002287Pricing Error -0.000346Pricing Error 0.001182Pvalue for Pricing Error 0.543151R2 0.122705

OLS Betas

Constant DOL IERE

beta -0.000625 -0.777264 0.500977se 0.000771 0.037033 0.060976tsts 0.000708 0.061093 0.067216beta 0.002099 1.668472 -0.450506se 0.002485 0.119302 0.196433tsts 0.002089 0.210809 0.252641beta -0.002100 1.155240 -0.016662se 0.001993 0.095684 0.157545tsts 0.001725 0.172851 0.215431beta -0.000783 0.732826 0.405030se 0.001225 0.058834 0.096871tsts 0.001209 0.069522 0.106408beta 0.000625 0.777264 0.499023se 0.000771 0.037033 0.060976tsts 0.000708 0.061093 0.067216

59

Table 26: Shanken OLS Results for Two Factor Model: DOL, IERE

Shanken OLS Pricing Errors

Statistic

Pricing Error p 0.000156Pricing Error p 0.001258Pricing Error p -0.002287Pricing Error p -0.000346Pricing Error p 0.001182Pvalue 0.552604R2 0.122705

Shanken OLS Lambdas

lambda se tsts

DOL 0.000594 0.002073 0.286438IERE 0.001915 0.001858 1.030357

Table 27: GMM Results for Two Factor Model: DOL, IERE

GMM b Vector

DOL IERE

beta -0.992274 12.008992se 4.925166 11.525998tsts -0.201470 1.041905beta -3.257403 21.006097se 4.409713 7.326131tsts -0.738688 2.867284

GMM Lambdas

DOL IERE

lambda 0.000594 0.001915se 0.002063 0.001789tsts 0.287814 1.069989lambda 0.000351 0.003217se 0.002187 0.001306tsts 0.160695 2.463973

GMM: Other Statistics

Iteration 1 Iteration 2

Jstat 2.192476 2.192476Degrees of Freedom 3.000000 3.000000Pvalue 0.533432 0.533432R-squared 0.122705 -0.099499

60

Table 28: OLS Results for Two Factor Model: DOL, DKP

OLS Lambdas

lambda se tsts

DOL 0.927710 0.924129 1.003875DKP -0.001415 0.001384 -1.022287

OLS Pricing Errors

Statistic

Pricing Error -0.004349Pricing Error 0.001126Pricing Error 0.001671Pricing Error 0.000975Pricing Error 0.002511Pvalue for Pricing Error 0.005793R2 0.694874

OLS Betas

Constant DOL DKP

beta -0.003753 0.000731 0.323874se 0.002215 0.001557 0.098077tsts 0.002184 0.001575 0.129424beta 0.001015 -0.000613 -0.060496se 0.002289 0.001609 0.101367tsts 0.002399 0.001521 0.108697beta 0.001577 -0.001832 -0.050557se 0.002839 0.001996 0.125737tsts 0.002634 0.001984 0.182540beta 0.000722 0.008989 -0.140004se 0.011761 0.008269 0.520857tsts 0.002586 0.009411 0.288661beta 0.003753 -0.000731 0.676126se 0.002215 0.001557 0.098077tsts 0.002184 0.001575 0.129424

61

Table 29: Shanken OLS Results for Two Factor Model: DOL, DKP

Shanken OLS Pricing Errors

Statistic

Pricing Error p -0.004349Pricing Error p 0.001126Pricing Error p 0.001671Pricing Error p 0.000975Pricing Error p 0.002511Pvalue 0.066339R2 0.694874

Shanken OLS Lambdas

lambda se tsts

DOL 0.927710 1.218457 0.761381DKP -0.001415 0.001484 -0.953123

Table 30: GMM Results for Two Factor Model: DOL, DKP

GMM b Vector

DOL DKP

beta 0.800243 -0.922490se 0.153861 5.922262tsts 5.201066 -0.155766beta 0.675880 -4.724938se 0.111092 5.317868tsts 6.083949 -0.888502

GMM Lambdas

DOL DKP

lambda 0.927710 -0.001415se 0.200208 0.001434tsts 4.633721 -0.986857lambda 0.789186 -0.002346se 0.147060 0.001356tsts 5.366414 -1.730170

GMM: Other Statistics

Iteration 1 Iteration 2

Jstat 7.122488 7.122488Degrees of Freedom 3.000000 3.000000Pvalue 0.068094 0.068094R-squared 0.694874 0.676755

62

Table 31: OLS Results for Two Factor Model: DOL, IMM

OLS Lambdas

lambda se tsts

DOL 0.003158 0.002387 1.323322IMM 0.007970 0.008983 0.887247

OLS Pricing Errors

Statistic

Pricing Error -0.000511Pricing Error -0.001128Pricing Error -0.000231Pricing Error 0.001157Pricing Error 0.000009Pvalue for Pricing Error 0.183793R2 0.955420

OLS Betas

Constant DOL IMM

beta -0.000691 -0.802515 0.168761se 0.000610 0.028652 0.007609tsts 0.000615 0.028595 0.009661beta -0.000850 1.170371 -0.215124se 0.000650 0.030555 0.008114tsts 0.000649 0.039716 0.007018beta 0.000037 1.136234 -0.212498se 0.000803 0.037740 0.010022tsts 0.000797 0.049369 0.008929beta 0.001397 1.007783 -0.184349se 0.000892 0.041911 0.011129tsts 0.000932 0.050665 0.009799beta 0.000691 0.802515 0.831239se 0.000610 0.028652 0.007609tsts 0.000615 0.028595 0.009661

63

Table 32: Shanken OLS Results for Two Factor Model: DOL, IMM

Shanken OLS Pricing Errors

Statistic

Pricing Error p -0.000511Pricing Error p -0.001128Pricing Error p -0.000231Pricing Error p 0.001157Pricing Error p 0.000009Pvalue 0.187354R2 0.955420

Shanken OLS Lambdas

lambda se tsts

DOL 0.003158 0.002387 1.323276IMM 0.007970 0.008983 0.887220

Table 33: GMM Results for Two Factor Model: DOL, IMM

GMM b Vector

DOL IMM

beta 3.428210 -0.175064se 3.544121 0.700366tsts 0.967295 -0.249961beta 3.328349 -0.169140se 3.519351 0.695659tsts 0.945728 -0.243136

GMM Lambdas

DOL IMM

lambda 0.003158 0.007970se 0.002376 0.008935tsts 1.329186 0.891962lambda 0.003069 0.007750se 0.002346 0.008743tsts 1.308472 0.886461

GMM: Other Statistics

Iteration 1 Iteration 2

Jstat 4.541785 4.541785Degrees of Freedom 3.000000 3.000000Pvalue 0.208593 0.208593R-squared 0.955420 0.954283

64

Table 34: OLS Results for Two Factor Model: DOL, IUERE

OLS Lambdas

lambda se tsts

DOL 0.000529 0.002071 0.255207IUERE 0.003651 0.001915 1.906757

OLS Pricing Errors

Statistic

Pricing Error 0.000524Pricing Error 0.001080Pricing Error 0.000126Pricing Error -0.001250Pricing Error 0.000761Pvalue for Pricing Error 0.632615R2 0.136338

OLS Betas

Constant DOL IUERE

beta -0.000228 -0.767972 0.538913se 0.000701 0.032149 0.041984tsts 0.000727 0.041882 0.056133beta 0.001482 0.950322 -0.255532se 0.001228 0.056307 0.073533tsts 0.001231 0.070996 0.072478beta 0.000147 1.478771 0.072929se 0.001995 0.091454 0.119433tsts 0.001863 0.158512 0.141746beta -0.001540 1.115343 0.292827se 0.001761 0.080726 0.105423tsts 0.002022 0.101674 0.127934beta 0.000228 0.767972 0.461087se 0.000701 0.032149 0.041984tsts 0.000727 0.041882 0.056133

65

Table 35: Shanken OLS Results for Two Factor Model: DOL, IUERE

Shanken OLS Pricing Errors

Statistic

Pricing Error p 0.000524Pricing Error p 0.001080Pricing Error p 0.000126Pricing Error p -0.001250Pricing Error p 0.000761Pvalue 0.651134R2 0.136338

Shanken OLS Lambdas

lambda se tsts

DOL 0.000529 0.002072 0.255106IUERE 0.003651 0.001931 1.891244

Table 36: GMM Results for Two Factor Model: DOL, IUERE

GMM b Vector

DOL IUERE

beta -0.871479 14.003768se 4.475836 7.827042tsts -0.194708 1.789152beta -1.152659 18.248721se 4.439239 6.229310tsts -0.259652 2.929493

GMM Lambdas

DOL IUERE

lambda 0.000529 0.003651se 0.002055 0.001974tsts 0.257238 1.849652lambda 0.000681 0.004757se 0.002126 0.001581tsts 0.320287 3.009187

GMM: Other Statistics

Iteration 1 Iteration 2

Jstat 1.402579 1.402579Degrees of Freedom 3.000000 3.000000Pvalue 0.704930 0.704930R-squared 0.136338 -0.098667

66

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