hw2_answer_econ120c_su07_S1.pdf

Economics 120C Name: _________________________ Professor Yongil Jeon Summer 2007-1 Student ID#: _________________________

Answer to Homework #2 – Summer 2007-Session 1, Econ 120C

(Final Exam, Summer 2006, Econ 120C) Answer all questions on separate paper. This problem set should be handed in to your TA at the beginning of your review session on Wednesday, August 1st, 2007. Problem sets may not be handed in once solutions have been distributed. Please write down your name and PID clearly. Good luck! (1-a) (3 points) Autoregressive distributed lag models include

a. current and lagged values of the error term. b. lags of the dependent variable, and lagged values of additional predictor

variables. c. current and lagged values of the residuals. d. lags and leads of the dependent variable.

Answer: b

(1-b) (3 points) The jth autocorrelation coefficient is defined as

a. 1

1

cov( , )var( ) var( )

t t

t t

Y YY Y

−

−

.

b. cov( , )

var( ) var( )t t j

t t j

Y XY X

−

−

.

c. cov( , )var( ) var( )

t t

t t

Y uY u

.

d. cov( , )

var( ) var( )t t j

t t j

Y YY Y

−

−

.

Answer: d

(1-c) (3 points) An autoregression is a regression

a. of a dependent variable on lags of regressors. b. that always allows for the errors to be correlated. c. model that relates a time series variable to its past values. d. to predict sales in a certain industry.

Answer: c

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2 Answer to HW #2, ECO 120C, SUMMER 2007 – Session 1

(1-d) (3 points) The times series regression with multiple predictors

a. is the same as the ADL(p,q) with additional predictors and their lags present.

b. gives you more than one prediction. c. cannot be estimated by OLS due to the presence of multiple lags. d. requires that the k regressors and the dependent variable have nonzero,

finite eighth moments.

Answer: a (1-e) (3 points) The Granger causality test

a. uses the F-statistic to test the hypothesis that certain regressors have no predictive content for the dependent variable beyond that contained in the other regressors.

b. establishes the direction of causality (as used in common parlance) between X and Y in addition to correlation.

c. is a rather complicated test for statistical independence. d. is a special case of the augmented Dickey-Fuller test.

Answer: a

(1-f) (3 points) In time series, the definition of causal effects

a. says that one variable helps predict another variable. b. does not make much sense since there are not multiple subjects. c. assumes that the same subject is being given different treatments at

different points in time. d. requires panel data.

Answer: c


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(1-h) (3 points) The concept of exogeneity is important because

a. it clarifies whether or not the variable is determined inside or outside your model.

b. maximum likelihood estimation is no longer valid. c. under strict exogeneity, OLS may not be efficient as an estimator of

dynamic causal effects. d. endogenous variables are not stationary, but exogenous are.

Answer: c

(1-i) (3 points) GLS is consistent and BLUE if

a. X is predetermined. b. the error process is AR(1). c. X is strictly exogenous. d. all the roots are inside the unit circle.

Answer: c

2. (6 points) Consider the following panel data regression with a single explanatory variable

Yit = β0 + β1Xit + uit.

In each of the examples below, you will be adding entity and time fixed effects. Indicate the total number of coefficients that need to be estimated.

(a) (3 points) The effect of saving rates on per capita income, data for three decades

(1960-1969, 1970-1979, 1980-1989; one observation per decade), 104 countries of the world.

Answer: 107 coefficients (2 time fixed effects, 103 entity fixed effects, intercept, slope).

(b) (3 points) The effect of pitching quality in baseball (as measured by the Team

ERA) on the winning percentage, annual data, 1998-1999 season, 1999-2000 season, 30 teams.

Answer: 32 coefficients (1 time fixed effect, 29 entity fixed effects, intercept, slope).


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3. (10 points) A study attempts to investigate the role of the various determinants of regional Canadian unemployment rates in order to get a better picture of Canadian aggregate unemployment rate behavior. The annual data (1967-1991) is for five regions (Atlantic region, Quebec, Ontario, Prairies, and British Columbia), and four age-gender groups (female and male, adult and young). Focusing on young females, the authors find significant effects for the following variables: the regional relative minimum wage rate (minimum wages divided by average hourly earnings), the regional share of youth in the labor force, the regional share of adult females in the labor force, United States activity shocks (deviations of United States GDP from trend), an indicator of the degree of monetary tightness in Canada, regional union density, and a regional index of unemployment insurance generosity. Explain why the authors only used region fixed effects. How would their specification have to change if they also employed time fixed effects? Answer: Since the study used Canada-wide effects (United States activity shocks, and

monetary tightness), these are identical for all regions at a point in time. Using time fixed effects in addition to these two variables would have generated perfect multicollinearity among the regressors, and hence the OLS estimator would not exist. An alternative specification would include time fixed effects, but eliminate variables which are constant across all regions at a given point in time.


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4. (15 points) Our class suggested an “entity-demeaned” procedure to avoid having to specify a potentially large number of binary variables. While it is somewhat tedious to specify a binary variable for each entity, this can still be handled relatively easily in the case of the 48 contiguous states. The idea of the “entity-demeaned” procedure was introduced as a computationally convenient and simplifying procedure. Now we use a “time-demeaned” procedure. Using the following equation

Yit = β0 + β1Xit + β3St + uit,

show how β1 can be estimated by the OLS regression using “time-demeaned” variables. Answer: Taking averages on both sides of the above equation results in

1 0 3 tt t tY X S uβ β β= + + +

where 1 1

1 1, ,n n

t it t iti i

Y Y X Xn n= =

= =∑ ∑ and 1

1 n

t iti

u un =

= ∑ .

Subtracting the averaged equation from the original one yields

1( ) ( )tit t it t itY Y X X u uβ− = − + − or ititit uXY += 1β where, titittitittitit uuuXXXYYY −=−=−= ,, . The “time-demeaned” regression can then be estimated by OLS.


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5. (15 points) Consider the model ttt uYY ++= −17.03.0 , where )1,0(~ iidut . Find the mean and variance of tY . Answer: Rewrite the AR(1) model as follows

122

121

7.07.0)7.01(3.0

)7.03.0(7.03.07.03.0

−−

−−−

++++=

++++=++=

ttt

tttttt

uuYuuYuYY

Continuing the substitution indefinitely then results in

.7.0)7.07.07.01(3.00

32 ∑∞

=−+++++=

iit

it uY

Given the result for the sum of a geometric series, the final expression is

.7.017.07.01

3.000∑∑∞

=−

∞

=− +=+

−=

iit

i

iit

it uuY

To find the mean and the variance, take first expectations on both sides

( ) ( ) 17.017.0100

=+=

+= ∑∑

∞

=−

∞

=−

iit

i

iit

it uEuEYE , since ( ) 0tE u = for all t.

To derive the variance, note that ( ) ∑∞

=−=−

0

7.0i

iti

tt uYEY . Hence the variance is

( )[ ] ( ) ( ) ( ) 2.040851.01

7.017.07.0 2

2

0

2

0

2222 ==−

===− ∑∑∞

=

∞

=−

u

i

i

iuit

itt uEYEYE σ

σ .

.


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6. (12 points) Decide if each of the following statements is true or false, and give a brief explanation of your decision:

(a) (3 points) Like cross-sectional observations, we can assume that most time series

observations are independently distributed.

Answer: Most time series processes are correlated over time, and many of them strongly correlated. This means they cannot be independent across observations, which simply represent different time periods.

(b) (3 points) The OLS estimator in a time series regression is unbiased without (i)

the homoskedasticity and (ii) no serial correlation assumptions.

Answer: Yes. It requires the assumptions of (i) linear in parameters, (2) Strict exogeneity assumption, (3) no perfect collinearity assumptions.

(c) (3 points) A trending variable cannot be used as the dependent variable in

multiple regression analysis.

Answer: Disagree. Trending variables are used all the time as dependent variables in a regression model. We do need to be careful in interpreting the results because we may simply find a spurious association between yt and trending explanatory variables. Including a trend in the regression is a good idea with trending dependent or independent variables.

(d) (3 points) Seasonality is not an issue when using annual time series observations. (Hint: Consider the definition of seasonality)

Answer: With annual data, each time period represents a year and it is not associated with any season.


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7. (10 points) The following model was estimated

174.0,554)013.0()042.0()045.0()042.0()56.0(

031.0073.0074.0344.054.1

2

1321

==

++++= −−−−

Rn

spipipipip ttttt

Where ip is the percentage change in monthly industrial production at an annualized rate, and sp is the percentage change in the Standard & Poor’s 500 Index at an annualized rate. (i) If the past three months of ip are zero and spt-1=0, what is the predicted

growth in industrial production for this month? Is it statistically different from zero?

Answer: This is given by the estimated intercept, 1.54. It is statistically different from zero since t=1.54/0.56=2.75.

(ii) If the past three months of ip are zero but spt-1=10, what is the predicted

growth in industrial production?

Answer: 1.54+0.031*10 = 1.85. Note that you could obtain the standard error of this estimate by running the regression: ip t on ipt-1, ipt-2, ipt-3, spt-1-10, and obtaining the standard error on the intercept.

(iii) What do you conclude about the effects of the stock market on real

economic activity? Answer: Growth in the S&P 500 index has a statistically significant effect on industrial production growth, in the Granger Causality sense, because the t statistic on spt-1 is about 2.38. The economic effect is reasonably large.


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8. (8 points) Money supply is linked to the monetary base by the money multiplier. Macroeconomic textbooks tell you that the central bank cannot control the money supply, but it can control the monetary base. As a result, you decide to specify a distributed lag equation of the growth in the money supply on the growth in the monetary base. One of your peers tells you that this is not a good idea for modeling the relationship between the two variables (thus, the outcomes are biased). What does she mean? Answer: Although the monetary base is one of the determinants of the money supply,

there are other factors, such as interest rates, that have an effect on the money multiplier. Hence there is the problem of omitted variables. If interest rates are correlated with the monetary base, then the OLS estimator will be inconsistent. Furthermore, it is likely that due to financial innovations, dynamic causal effects have changed over time. Finally there is the concern of simultaneous causality bias. If the Federal Reserve changes the monetary base as a result of changes in the money supply, perhaps as a result of targeting, then the monetary base becomes endogenous.


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hw2_answer_econ120c_su07_S1.pdf