CHAPTER 5 INTERNATIONAL PARITY RELATIONSHIPS AND FORECASTING

FOREIGN EXCHANGE RELATIONSHIPS

SUGGESTED ANSWERS AND SOLUTIONS TO END-OF-CHAPTER

QUESTIONS AND PROBLEMS

QUESTIONS

1. Give a full definition of arbitrage.

Answer: Arbitrage involves the simultaneous purchase and sale of the same or equivalent assets or

commodities – buying at a low price, selling at a high price – thus making a riskless profit.

2. Discuss the implications of the interest rate parity for the exchange rate determination.

Answer: Assuming the forward exchange rate is an unbiased predictor of the future spot rate, IRP

between dollars and pounds can be written as:

S = [(1 + i£)/(1 + i$)]E[St+1|It].

The exchange rate is thus determined by relative interest rates and the expected future spot rate

conditional on available information, It, as of the present time. The expectation is self-fulfilling.

Since the information set is continuously updated as news hit the market, the exchange rate

exhibits a dynamic, random behavior.

4. Explain purchasing power parity, both the absolute and relative versions. What causes deviations

from the purchasing power parity?

Answer: Absolute (or static) purchasing power parity (PPP) is S = P$/P£ where S is the spot

exchange rate, P$ is the price level in the dollar country (Canada) and P£ is the price level in the

pound country (Britain). Relative (or dynamic) PPP is e = π$ - π£ where e is the rate of change of

the spot exchange rate, π$ is the expected change of price level (expected inflation) in the dollar

country (Canada) and π£.is the expected change of price level (expected inflation) in the pound

country (Britain).

PPP can be violated by barriers to trade – such as tariffs or transportation costs – or cross-country

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differences in tastes. PPP is the law of one price applied to a standard consumption basket.

5. Discuss the implications of the deviations from purchasing power parity for countries’ competitive

positions in the world market.

Answer: If exchange rate changes satisfy PPP, the competitive positions of countries remain

unaffected by the exchange rate changes. Such exchange rate changes are strictly “nominal”.

Otherwise, exchange rate changes affect relative the competitiveness of industries in the countries

involved. If a country’s currency appreciates (depreciates) by more than is warranted by PPP, that

country’s real exchange rate rises. The effect is to raise the relative price of that country’s

exported goods while reducing (for domestic use or consumption) the relative price of that

country’s imports. Exporters are hurt. Importers are helped.

6. Explain and derive the International Fisher Effect.

Answer: The International Fisher Effect combines domestic Fisher effects in two countries and the

relative version of PPP in its expectational form. Specifically, the country-specific Fisher effect

for a dollar country and a pound country is, respectively …

E(π$) = i$ - ρ$ where E(π$) is the expected rate of inflation, i$ is the nominal interest rate

and ρ$ is the real rate of interest in the dollar country, and likewise,

E(π£) = i£ - ρ£. in the pound country.

Assuming the real interest rate is the same in the two countries, i.e., ρ$ = ρ£, and substituting the

above results into the PPP, i.e., E(e) = E(π$)- E(π£), the international Fisher effect is: E(e) = i$ - i£.

7. Researchers found that it is very difficult to forecast future exchange rates more accurately than the

forward exchange rate or the current spot exchange rate. How would you interpret this finding?

Answer: Foreign exchange markets are informationally efficient. Unless one has private

information that is not yet reflected in the current market rates, which is the case for virtually all

participants in foreign exchange markets, all new relevant information is truly new. New

information that can either raise the exchange rate or lower it is equally likely to raise or lower the

exchange rate. The random flow of new information generates random movements of the exchange

rate.

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8. Explain the random walk model for exchange rate forecasting. Can itbe consistent with the

technical analysis?

Answer: If exchange rates follow a random walk, the current exchange rate is the best predictor of

the future exchange rate. In that case the past history of the exchange rate is of no value in

predicting future exchange rate. A random walk model is inconsistent with technical analysis

which tries to use past history to predict future exchange rate.

9. Derive and explain the monetary approach to exchange rate determination.

Answer: The monetary approach to exchange rate movements is based on two tenets: purchasing

power parity and the quantity theory of money. Combining these two theories suggests the $/£ spot

exchange rate as: S($/£) = (M$/M£)(V$/V£)(y£/y$),

where M denotes the rate money supply, V is the velocity of money (essentially national income

divided by money supply) and y is national income. What matters in exchange rate determination are:

i. relative money supply,

ii. relative velocities of money, and

iii. relative national incomes.

10. [CFA question: 1997, Level 3.] Explain the following three concepts of purchasing power parity

(PPP):

a. The law of one price

b b. Absolute PPP

c. Relative PPP

Answer:

a. The Law of One Price (LOP) maintains that the same good (or basket of goods) must have the

same price in two places. Otherwise, arbitrage – arbitragers buying at the low-priced site and

selling at the higher priced site – will ensue until the prices are the same in the two sites.

b. Absolute Purchasing Power Parity (absolute PPP) holds that the price level in a country is equal

to the price level in another country times the exchange rate between the two countries. i.e., P 1/P2

= S1/2 There is virtually no empirical evidence to support Absolute Purchasing Power Parity.

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c. Relative Purchasing Power Parity (relative PPP) holds that the rate of exchange rate change

between a pair of countries is equal to the difference in expected inflation rates between the two

countries.

11. [CFA question: 1997, Level 3.] Evaluate the usefulness of relative PPP in predicting movements in

foreign exchange rates on:

a. Short-term basis (for example, three months)

b. Long-term basis (for example, six years)

Answer:

a. PPP is not useful for predicting exchange rates on the short-term basis mainly because

international commodity arbitrage is a costly process.

b. PPP is at most useful for predicting exchange rates on the long-term basis.

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PROBLEMS

1. Suppose that the treasurer of Weston’s has an extra cash reserve of $10 million to invest for six

months. The six-month interest rate is 4 percent per annum in Canada and 5 percent per annum in

Germany. Currently, the spot exchange rate is €0.65 per dollar and the six-month forward

exchange rate is €0.66 per dollar. The treasurer of Weston’s does not wish to bear any exchange

risk. Where should he invest?

Solution: The market conditions are summarized as follows:

i$ = 4%; i€ = 5%; S = €0.65/$; F = €0.66/$.

If $10 million is invested in Canada, the maturity value in six months will be

$10,200,000 = $10,000,000 (1 + (.04/2)).

Alternatively, $10 million can be converted into euro and invested at the European interest rate,

with the euro maturity value sold forward for forward cover. In this case the dollar maturity value

will be … $10,094,700 = ($10,000,000 x 0.65)(1 + (.05/2))(1/0.66)

Clearly it is better to invest $10 million in Canada. Even though the European interest rate exceeds

the interest rate in Canada, the expected depreciation of the euro vis-à-vis the Canadian dollar that

is built into the relation between the spot and forward rates more than offsets the international

interest differential.

2. While you were visiting Paris you purchased a Renault for €10,000 payable in three months. You

have enough cash at your bank in Vancouver, earning interest of 0.35 percent per month

compounded monthly, to pay for the car. Currently, the spot exchange rate is $1.45/€ and the 3-

month forward exchange rate is $1.40/€. In Paris the money market interest rate is 2.0 percent for

a 3-month investment.

There are two alternative ways of paying for your Renault.

a. Keep the funds at your bank in Canada and buy €10,000 forward.

b. Buy a certain euro amount spot today and invest the amount in Europe for three months so that

the maturity value become equal to €10,000. Evaluate each payment method. Which method would

you prefer? Why?

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Solution: The problem situation is summarized as follows:

Obligation = €10,000 payable in 3 months

i$ = 0.35 percent per month compounded monthly

i€ = 2.0 percent for 3 months

S = $1.45/€

F = $1.40/€

Option a:

To buy €10,000 forward, you will need 10,000*1.40 or $14,000 in 3 months to fulfill the forward

contract. The present value of $14,000 discounted at the dollar interest rate is:

$14,000/(1.0035)3 = $13,854

Thus, the cost of the car as of today is $13,854.

Option b:

The present value of €10,000 is €9,804 = €10,000/(1.02). To buy €9,804 today will cost $14,216

= $9,804 x1.45. Thus the dollar cost of the Renault as of today is $14,216.

Option “a” is preferred. The difference between Options “a” and “b” is $362, or $14,216 minus

$13,854.

3. Currently, the spot exchange rate is $1.50/€ and the 3-month forward exchange rate is $1.49/€. The

3-month interest rate is 4 percent per annum in Canada and 5 percent per annum in Europe.

Assume you can borrow as much as $1,500,000 or €1,000,000.

a. Determine whether interest rate parity is currently holding.

a. If IRP is not holding, how would you carry out covered interest arbitrage? Show all the

steps and determine the arbitrage profit.

c. Explain how IRP will be restored as a result of covered arbitrage activities.

Solution: First, summarize the given data:

S = $1.50/€; F = $1.49/€; i$ = 4 %; i€ = 5 %

Credit = $1,500,000 or €1,000,000

a. In Canada investment: (1+i$) = (1 + (0.04/4)) = 1.0100

Covered investment in Europe: (1+i€)(F/S) = (1 + (0.05/4))*(1.49/1.50) = 1.00575

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Thus, IRP is not holding exactly.

The arbitrager will borrow in Euro and invest in Canadian dollar securities.

b. 1. Borrow €1,000,000; repayment will be €1,012,500 = €1,000,000 *(1.0125)

2. Buy $1,500,000 spot using €1,000,000.

3. Invest $1,500,000 at $ interest rate; maturity value will be $1,515,000

4. Buy €1,012,500 forward for $1,508,625 = €1,012,500 * 1.49

Arbitrage profit will be $6,375.

c. In the process of the arbitrage transactions described above,

The dollar interest rate will fall. ( Canadian securities prices rise. )

The euro interest rate will rise. ( European securities prices fall. )

The spot exchange rate (in $/€ ) will fall. ( Up from S = $1.50/ € )

The forward exchange rate (in $/€ ) will rise. ( Down from F = $1.49/€ )

These adjustments continue until IRP holds.

4. Suppose that the current spot exchange rate is €0.65/$ and the three-month forward exchange rate

is €0.64/$. The three-month interest rate is 5.6 percent per annum in Canada and 5.4 percent per

annum in France. Assume that you can borrow up to $1,000,000 or €1,060,000.

a. Show how to realize a certain profit via covered interest arbitrage, assuming that you

want to realize profit in terms of dollars. Also determine the size of your arbitrage

profit.

b. Assume that you want to realize profit in terms of euros. Show the covered arbitrage

process and determine the arbitrage profit in euros.

Solution: The market data are summarized as follows:

S = €0.65/$ = $1.5385/€;

F = €0.64/$ = $1.5625/€;

On a 3-month basis: i$ = (0.056/4) = 0.014; i€ = (0.054/4) = 0.0135

(1+i$) = 1.014 < (1+i€)(F/S) = (1.0135)*(0.65/0.64) = 1.029336

a. 1. Borrow $1,000,000; repayment will be $1,014,000

2. Buy €650,000 spot for $1,000,000.

3. Invest in France at 5.4 % for 3 months; maturity value will be €658,775

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4. Sell €658,775 forward for $1,029,336 = €658,775* ($1.5625/€)

Arbitrage profit will be $15,336 = $1,029,336 - $1,014,000

b. 1. Borrow $1,000,000; repayment will be $1,014,000.

2. Buy €650,000 spot for $1,000,000.

3. Invest in France at 5.4 % for 3 months; maturity value will be €658,775

4. Buy $1,014,000 forward for €648,960 = $1,014,000 / ($1.5625/€)

Arbitrage profit will be €9,815 = €658,775 - €648,960

Note that only step (4) is different.

5. The Economist reports that the interest rate per annum is 5 percent in Canada and 50 percent in

Turkey. Why do you think the interest rate is so high in Turkey? On the basis of the reported

interest rates, how would you predict the change of the exchange rate between the Canadian

dollar and the Turkish lira?

Solution: The high Turkish interest rate reflects high expected inflation in Turkey. According to

international Fisher effect (IFE), we have

E(e) = i$ - iLira

= 5 - 50 = - 45 %

The Turkish lira thus is expected to depreciate against the Canadian dollar at an annual rate of

approximately 45 percent per annum.

6. As of November 1, 2007, the exchange rate between the Brazilian real and US dollar is R$2.10/$.

The consensus forecast for the US and Brazil inflation rates for the next one year period is 3.00

percent and 15 percent, respectively. How would you forecast the exchange rate to be at around

November 1, 2008?

Solution:

In view of the significant difference in inflation rates between the US and Brazil, we may invoke

purchasing power parity to forecast the exchange rate.

E(e) = E(π$) - E(πR$)

= 3.0% - 15.0%

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= -12.0%

E(ST) = So(1 + E(e))

= (R$2.10/$) (1 + 0.12)

= R$2.35/$

7. Omni Advisors, an international pension fund manager, uses the concepts of purchasing power parity (PPP) and the International Fisher Effect (IFE) to forecast spot exchange rates. Omni gathers the financial information as follows:

Base price level 100

Current Canadian price level 105

Current South African price level 111

Base rand spot exchange rate $0.175

Current rand spot exchange rate $0.158

Expected annual Canadian inflation 7 %

Expected annual South African inflation 5 %

Expected Canadian one-year interest rate 10 %

Expected South African one-year interest rate 8 %

Calculate the following exchange rates (ZAR refers to the South African rand):

a. The current ZAR spot rate in Canadian dollars that would have been forecast by PPP.

b. Using IFE, the expected ZAR spot rate in dollars one year from now.

c. Using PPP, the expected ZAR spot rate in dollars four years from now.

Solutions:

a. ZAR spot rate under PPP = [1.05/1.11](0.175) = $0.1655/ZAR.

b. Expected ZAR spot rate = [1.10/1.08] (0.158) = $0.1609/ZAR.

c. Expected ZAR spot under PPP = [(1.07)4/(1.05)4] (0.158) = $0.1704/ZAR.

8. Suppose that the current spot exchange rate is €1.50/₤ and the one-year forward exchange rate is

€1.60/₤. The one-year interest rate is 5.4 percent in euros and 5.2 percent in pounds. You can

borrow at most €1,000,000 or the equivalent pound amount, that is, ₤666,667 at the current spot

exchange rate.

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a. Show how you can realize a guaranteed profit from covered interest arbitrage. Assume that you are a euro-based investor. Also determine the size of the arbitrage profit.

b. Discuss how interest rate parity may be restored as a result of the above transactions. c. Suppose you are a pound-based investor. Show the covered arbitrage process and

determine the pound profit amount.

Solutions:

a. First, note that (1+i €) = 1.054 is less than (F/S)(1+i €) = (1.60/1.50)*(1.052) = 1.1221.

You should thus borrow in euros and lend in pounds. The steps are …

1. Borrow €1,000,000 and promise to repay €1,054,000 in one year.2. Buy ₤666,667 spot for €1,000,000.3. Invest ₤666,667 at the pound interest rate of 5.2 percent; maturity value is ₤701,334.4. To hedge exchange risk, sell the maturity value ₤701,334 forward in exchange for €1,122,134.

Arbitrage profit is €1,122,134 minus €1,054,000, i.e., €68,134.

b. As a result of the above arbitrage transactions, capital flows from the euro to the pound causing the euro interest rate to rise and the pound interest rate to fall. In addition, because of spot buying pressure on the euro (selling pressure on the pound) the spot exchange rate (€/₤) will rise. Because of the reverse covering forward transactions, the forward rate (€/₤) will fall. These adjustments continue until interest rate parity is restored.

c. The pound-based investor will carry out the same transactions 1, 2 and 3 in the solution to Part “a”. To hedge, she will buy €1,054,000 forward in exchange for ₤658,750. The arbitrage profit in sterling will then be ₤701,334 minus ₤658,750, i.e., ₤42,584 .

9. Due to the integrated nature of their capital markets, investors in both Canada and the United Kingdom require the same real interest rate, 2.5 percent, on their lending. There is a consensus in capital markets that the annual inflation rate is likely to be 3.5 percent in Canada and 1.5 percent in the United Kingdom for the next three years. The spot exchange rate is currently $2.30/£.

a. Compute the nominal interest rate per annum in both Canada and the United Kingdom, assuming that the Fisher effect holds.

b. What is your expected future spot dollar-pound exchange rate three years from now?c. Can you infer the forward dollar-pound exchange rate for one-year maturity?

Solutions:

a. Nominal rate in Canada = (1+ρ) (1+E(π$)) – 1 = (1.025)(1.035) – 1 = 0.0609 or 6.09%. Nominal rate in UK = (1+ρ) (1+E(π₤)) – 1 = (1.025)(1.015) – 1 = 0.0404 or 4.04%.

b. E(ST) = [(1.0609)3/(1.0404)3] (2.30) = $2.4387/₤

c. F = [1.0609/1.0404](2.30) = $2.3462/₤

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Problems 10 to 15 are based on the following information on spot and forward foreign exchange quotes for the Canadian dollar against four major currencies recorded for 26 June 2007.

SPOT 1 mo 3 mo 6 mo 1 yr 2 yr 3 yr 4 yr 5 yr US$ 0.9340 0.9343 0.9350 0.9370 0.9418 0.9501 0.9569 0.9608 0.9603Euro 1.4411 1.4414 1.4423 1.4435 1.4454 1.4563 1.4646 1.4751 1.4961Yen 115.42 115.20 114.78 114.18 112.89 109.96 107.04 103.99 100.76£ 2.1382 2.1343 2.1259 2.1131 2.0902 2.0570 2.0339 2.0148 2.0128

10. Spot and forward exchange quotes for the Canadian dollar versus four major currencies are recorded above for 26 June 2007.

US$ and Yen are direct quotes - foreign currency units per Canadian dollar.Euro and British pound are indirect quotes - Canadian dollars per unit of foreign currency.

a. According to the forward rate structure, which currencies are expected to appreciate against the Canadian dollar over the next year? Explain briefly.

b. According to the forward rate structure, which currencies are expected to depreciate against the Canadian dollar over the next year? Explain briefly.

c. For each case, express the expected currency appreciation (or depreciation) in annual terms.

Solutions:

Expected to appreciate against Canadian dollar… Euro and YenEuro (direct quote) … further into the future, increasing amounts of Canadian dollars are required to by one Euro.Yen (indirect quote) … further into the future, fewer yen are required to buy one Canadian dollar.

Expected to depreciate against Canadian dollar… US dollar and British poundUS dollar (indirect quote) … further into the future, increasing amounts of US dollars are required to buy one Canadian dollar. British pound (direct quote) … further into the future, decreasing amounts of Canadian dollars are required to by one British pound.

Expected annual rate of appreciation, outward to one year …Euro 0.2984Yen 2.1920

Expected annual rate of depreciation, outward to one year …US$ -0.8351Pound -2.2449

Expected average annual rate of appreciation, outward to five years …Euro 0.9408Yen 3.3389

Expected average annual rate of depreciation, outward to five years …US$ -0.6966Pound -1.4544

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11. On the same day as mentioned in Problem 10, the yield on Canadian six-month Treasury bills (annual terms) was 4.71%. The corresponding yield on US six-month Treasury bills (in US dollars) was 4.96%.

Does interest rate parity hold between Canada and the US in view of the difference in six-month Treasury bill rates? Explain the difference, if any.

Solution:

Interest Rate Parity requires …

rUS - rCAN = rate of US$ depreciation = (F/S) -1

Treasury Bill Rates ( 6-month TB rate, not annualized )rCAN ... 6 months 0.023279 = 2.328 %rUS … 6 months 0.024500 = 2.450 %rUS - rCAN … 6 months 0.001221 = 12.2 bps

Expected US$ 6-month depreciation = (F/S) - 16-months 0.003212 = 32.1 bps

According to the 6-month TB rate differential, interest rate parity does not hold … 12.21 bps ≠ 32.12 bps

12. Where should an investor invest, Canada or the United States? Explain.

Solution:

Invest $1,000 in US 6-month Treasury Bills with forward cover …= 1,000 x (1+rUS) x (F/S)= 1,000 x (1+[(1.0496)^.5)-1]) x (1/0.9370)/(1/0.934)= 1,021.22

Invest $1,000 in Canada 6-month Treasury Bills …= 1,000 x (1+rCAN)= 1,000 x (1+ [(1.0471)^.5)-1])= 1,023.28

Therefore, invest in Canada.

Explanation:The expected rate of US dollar depreciation over the next six months [(F/S)-1] exceeds the six-month interest differential on US and Canadian Treasury Bills.

13. Assume that US-Canada six-month interest rate parity holds on a net-of-cost basis.What does your analysis imply about the cost of obtaining forward cover?

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Solution:

In Canadian dollars, the 6-month yield on the Canadian TB investment is 2.328 percent.In Canadian dollars, the 6-month yield on the US TB investment is 2.122 percent.

The difference, 20.6 basis points, represents the transactions cost in 6-month covered interest arbitrage.

14. Record the accuracy of the forward rates on 26 June 2007 as a predictor of the spot rates for the most recent relevant date for the reader today. For instance, if today is 1 July 2008, the (one year) forward rates observed on 26 June 2007 imply the following expected exchange rates on 26 June 2008:

US$ 0.9418Euro 1.4454Yen 112.89£ 2.0902

Find the spot rates on 26 June 2008 (or closest) and compare to the one-year forward rates observed on 26 June 2007. Compute the errors vis-à-vis the (forward rate) forecast implied by the forward rates observed on 26 June 2007.

(The results of this exercise depend on time-specific data collected to answer the questions.)

15. The one year Canadian Treasury bill rate is 4.71 percent. In view of the observed spot and one-year forward foreign exchange rates for the euro, British pound and Japanese yen, what are the respectively the interest-rate-parity-satisfying one-year Treasury bill rates in Europe, Britain and Japan? Ignore transactions costs.

Solution:

Interest Rate Parity requires …

… with indirect quotes for the SPOT and FORWARD exchange rates on the Canadian dollar (units of foreign currency per one Canadian dollar) …

1 + rCAN = (1 + rFOREIGN)*(F/S)

rFOREIGN = [(1 + rCAN) *(S/F)] - 1

… with direct quotes for the SPOT and FORWARD exchange rates on the Canadian dollar (Canadian dollars per unit is foreign currency) …

1 + rCAN = (1 + rFOREIGN)*((1/F)/(1/S))

rFOREIGN = [(1 + rCAN) *((1/S/(1/F))] - 1

The SPOT and FORWARD quotes for the Japanese yen are INDIRECT.The SPOT and FORWARD quotes for the Euro and British pound are DIRECT.

Solution values for interest-rate-parity satisfying country-specific one-year Treasury Bill rates …

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For Japanese one-year Treasury Bills …

r¥ = {1.0471 x [(112.89)/(115.42)]} -1= 0.024147626= 2.41 percent

For Euro one-year Treasury Bills …

r€ = {1.0471 x [(1/1.4454)/(1/1.4411)]} -1= 0.043984925= 4.40 percent

For British one-year Treasury Bills …

r£ = {1.0471 x [(1/2.0902)/(1/2.1382)]} -1= 0.071145929= 7.11 percent

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Mini Case: Turkish Lira and the Purchasing Power Parity

Refer to Chapter 5 pg. 122 in text.

Solution:

a. In the current solution, we use the monthly data from January 1999 – December 2002.

Turkey vs. U.S.

-0.05

0.00

0.05

0.10

0.15

-0.05 0 0.05 0.1 0.15

Inf_Turkey - Inf_US

Rat

e o

f C

hang

e of

TL/

$ (e

)

b. We regress exchange rate changes (e) on the inflation rate differential and estimate the intercept (α ) and slope coefficient (β):

3.095) (t 1.472 β̂

0.649)- (t 0.011 α̂

ε Inf_US) -Inf_Turkey (β̂ α̂ e tt

==

=−=++=

The estimated intercept is insignificantly different from zero, whereas the slope coefficient is positive and significantly different from zero. In fact, the slope coefficient is insignificantly different from unity. [Note that t-statistics for β = 1 is 0.992 = (1.472 – 1)/0.476 where standard error is 0.476]. In other words, we cannot reject the hypothesis that the intercept is zero and the slope coefficient is one. The results are thus supportive of purchasing power parity.

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1.47β̂ =

0.011α̂ −=