Introduction to Econometrics, 3e (Stock)
Chapter 15 Estimation of Dynamic Causal Effects
15.1 Multiple Choice
1) A distributed lag regression
A) is also called AR(p).
B) can also be used with cross–sectional data.
C) gives estimates of dynamic causal effects.
D) is sometimes referred to as ADL.
2) Heteroskedasticity– and autocorrelation–consistent standard errors
A) result in the OLS estimator being BLUE.
B) should be used when errors are autocorrelated.
C) are calculated when using the Cochrane–Orcutt iterative procedure.
D) have the same formula as the heteroskedasticity robust standard errors in cross–sections.
3) Sensitivity analysis of the results may include the following with the exception of
A) stability over time analysis of the estimated multipliers.
B) using homoskedasticity only rather than HAC standard errors.
C) investigation of omitted variable bias.
D) looking at different computations of the HAC standard errors.
4) A seasonal binary (or indicator or dummy) variable, in the case of monthly data,
A) is a binary variable that take on the value of 1 for a given month and is 0 otherwise.
B) is a variable that has values of 1 to 12 in a given year.
C) is a variable that contains 1s during a given year and is 0 otherwise.
D) does not exist, since a month is not a season.
5) Ascertaining whether or not a regressor is strictly exogenous or exogenous ultimately requires all of
the following with the exception of
A) economic theory.
B) institutional knowledge.
C) expert judgment.
D) use of HAC standard errors.
6) 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.
7) The distributed lag model is given by
A) Yt = β0 + β1Xt + β2Yt–1 + ut.
B) Yt = β0 + β1Yt–1 + β2Yt–2 + … + βrYt–r + ut.
C) Yt = β0 + β1ut + β2ut+1 + β3ut+2 + … + βr+1ut+r + et.
D) Yt = β0 + β1Xt + β2Xt–1 + β3Xt–2 + … + βr+1Xt–r + ut.
8) 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B) maximum likelihood estimation is no longer valid.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 variables are.
9) The impact effect is the
A) zero period dynamic multiplier.
B) h period dynamic multiplier, h>0.
C) cumulative dynamic multiplier.
D) long–run cumulative dynamic multiplier.
10) Estimation of dynamic multipliers under strict exogeneity should be done by
A) instrumental variable methods.
B) OLS.
C) feasible GLS.
D) analyzing the stationarity of the multipliers.
11) Autocorrelation of the error terms
A) makes it impossible to calculate homoskedasticity only standard errors.
B) causes OLS to be no longer consistent.
C) causes the usual OLS standard errors to be inconsistent.
D) results in OLS being biased.
12) The long–run cumulative dynamic multiplier
A) cannot be calculated since in the long–run, we are all dead.
B) is the sum of all individual dynamic multipliers.
C) is the coefficient on Xt–r in the standard formulation of the distributed lag model.
D) is the difference between the coefficient on Xt–1 and Xt–r.
13) The concepts of exogeneity, strict exogeneity, and predeterminedness
A) are defined in such a way that strict exogeneity implies exogeneity.
B) can be used interchangeably.
C) are defined in such a way that exogeneity implies strict exogeneity.
D) correspond to endogeneity, strict endogeneity, and lagged endogenous variables.
14) GLS
A) results in smaller variances of the estimator than OLS if the regressors are strictly exogenous.
B) is the same as OLS using HAC standard errors.
C) can be used even if the regressors are not strictly exogenous.
D) can be used for time–series estimation, but not in cross–sectional data.
15) Quasi differences in Yt are defined as
A) Yt – Yt–1.
B) Yt – φ1Yt–1.
C)
△
Yt – φ1Yt–1.
D) φ1(Yt – Yt–1).
16) Infeasible GLS
A) requires too much memory even for today’s PCs.
B) uses complicated interative techniques.
C) cannot be calculated since it also uses quasi differences for Xt.
D) assumes the parameters of the error autocorrelation process to be known.
17) The 95% confidence interval for the dynamic multipliers should be computed by using the estimated
coefficient ±
A) 1.96 times the RMSFE.
B) 1.96 times the HAC standard errors.
C) 1.96, since the HAC errors are standardized.
D) 1.64 times the HAC standard errors since the alternative hypothesis is one–sided.
18) The Cochrane–Orcutt iterative method is
A) a special case of GLS estimation.
B) a method to compute HAC standard errors.
C) a special case of maximum likelihood estimation.
D) a grid search for the autoregressive parameters on the error process.
19) To convey information about the dynamic multipliers more effectively, you should
A) plot them.
B) discuss these carefully one at a time.
C) estimate them by maximum likelihood methods.
D) first make sure that they are stationary.
20) GLS involves
A) writing the model in differences and estimating it by OLS, using HAC standard errors.
B) truncating the sample at both ends of the period, then estimating by OLS using HAC standard errors.
C) checking the AIC rather than the BIC in choosing the maximum lag–length of the regressors.
D) transforming the regression model so that the errors are homoskedastic and serially uncorrelated, and
then estimating the transformed regression model by OLS.
21) 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.
22) The distributed lag model assumptions include all of the following with the exception of:
A) There is no perfect multicollinearity.
B) Xt is strictly exogenous.
C) E(ut Xt, Xt–1, Xt–2) = 0
D) The random variables Xt and Yt have a stationary distribution.
23) In the distributed lag model, the coefficient on the contemporaneous value of the regressor is called
the
A) dynamic effect.
B) cumulative multiplier.
C) autoregressive error.
D) impact effect.
24) In the distributed lag model, the dynamic causal effect
A) is the sequence of coefficients on the current and lagged values of X.
B) is not the same as the dynamic multiplier.
C) is generated by choosing different truncation points for the HAC standard errors.
D) requires estimation of the model by Cochrane–Orcutt method.
25) HAC standard errors should be used because
A) they are convenient simplifications of the heteroskedasticity–robust standard errors.
B) conventional standard errors may result in misleading inference.
C) they are easier to calculate than the heteroskedasticity–robust standard errors and yet still allow you to
perform inference correctly.
D) when there is a structural break, then conventional standard errors result in misleading inference.
26) The interpretation of the coefficients in a distributed lag regression as causal dynamic effects hinges
on
A) the assumption that X is exogenous
B) not having more than four lags when using quarterly data
C) using GLS rather than OLS
D) the use of monthly rather than annual data
27) Given the relationship between the two variables, the following is most likely to be exogenous:
A) the inflation rate and the short term interest rate: short–term interest rate is exogenous
B) U.S. rate of inflation and increases in oil prices: oil prices are exgoneous
C) Australian exports and U.S. aggregate income: U.S. aggregate income is exogenous
D) change in inflation, lagged changes of inflation, and lags of unemployment: lags of unemployment are
exogenous
28) When Xt is strictly exogenous, the following estimator(s) of dynamic causal effects are available:
A) estimating an ADL model and calculating the dynamic multipliers from the estimated ADL
coefficients
B) using GLS to estimate the coefficients of the distributed lag model
C) neither (a) or (b)
D) (a) and (b)
29) In time series data, it is useful to think of a randomized controlled experiment
A) consisting of the same subject being given different treatments at different points in time
B) consisting of different subjects being given the same treatment at the same point in time
C) as being non–existent (this is a time series after all, and there are no real “parallel universes”
D) consisting of the at least two subjects being given different treatments at the same point in time
30) Consider the distributed lag model Yt = β0 + β1Xt + β2Xt–1 + β3Xt–2 + … + βr+1Xt–r + ut. The dynamic
causal effect is
A) β0 + β1
B) β1 + β2+…+βr+1
C) β0 + β1+…+βr+1
D) β1
15.2 Essays and Longer Questions
1) To estimate dynamic causal effects, your textbook presents the distributed lag regression model, the
autoregressive distributed lag model, and a quasi–difference representation of the distributed lag model
with autoregressive errors. Using a simple example, such as a distributed lag model with only the current
and past value of X and an AR(1) model for the error term, discuss how these models are related. In each
case suggest estimation methods and evaluate the relative merit in using one rather than the other.
2) Your textbook presents as an example of a distributed lag regression the effect of the weather on the
price of orange juice. The authors mention U.S. income and Australian exports, oil prices and inflation,
monetary policy and inflation, and the Phillips curve as other candidates for distributed lag regression.
Briefly discuss whether or not the exogeneity assumption is likely to hold in each of these cases. Explain
why it is so hard to come up with good examples of distributed lag regressions in economics.
3) 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. What does she mean?
4) In your intermediate macroeconomics course, government expenditures and the money supply were
treated as exogenous, in the sense that the variables could be changed to conduct economic policy to
influence target variables, but that these variables would not react to changes in the economy as a result
of some fixed rule. The St. Louis Model, proposed by two researchers at the Federal Reserve in St. Louis,
used this idea to test whether monetary policy or fiscal policy was more effective in influencing output
behavior. Although there were various versions of this model, the basic specification was of the following
type:
Δln(Yt) = β0 + β1Δln mt + … + βpΔln mt–p–1 + βp+1Δln Gt + … + βp+qΔln Gt–q–1 + ut
Assuming that money supply and government expenditures are exogenous, how would you estimate
dynamic causal effects? Why do you think this type of model is no longer used by most to calculate fiscal
and monetary multipliers?
5) Your textbook mentions heteroskedasticity– and autocorrelation– consistent standard errors. Explain
why you should use this option in your regression package when estimating the distributed lag
regression model. What are the properties of the OLS estimator in the presence of heteroskedasticity and
autocorrelation in the error terms? Explain why it is likely to find autocorrelation in time series data. If
the errors are autocorrelated, then why not simply adjust for autocorrelation by using some non–linear
estimation method such as Cochrane–Orcutt?
6) Your textbook presents as an example of a distributed lag regression the effect of the weather on the
price of orange juice. The authors mention U.S. income and Australian exports, oil prices and inflation,
monetary policy and inflation, and the Phillips curve as other potential candidates for distributed lag
regression. You are considering estimating the effect of minimum wages on teenage employment
(employment population ratio) using a time series of U.S. data. Write a short essay on whether a
distributed lag model would be a suitable tool to figure out dynamic causal effects in this case.
15.3 Mathematical and Graphical Problems
1) One of the central predictions of neo–classical macroeconomic growth theory is that an increase in the
growth rate of the population causes at first a decline the growth rate of real output per capita, but that
subsequently the growth rate returns to its natural level, itself determined by the rate of technological
innovation. The intuition is that, if the growth rate of the workforce increases, then more has to be saved
to provide the new workers with physical capital. However, accumulating capital takes time, so that
output per capita falls in the short run.
Under the assumption that population growth is exogenous, a number of regressions of the growth rate
of output per capita on current and lagged population growth were performed, as reported below. (A
constant was included in the regressions but is not reported. HAC standard errors are in brackets. BIC is
listed at the bottom of the table).
Regression of Growth Rate of Real Per–Capita GDP on Lags of Population Growth,
United States, 1825–2000
(1)
(2)
(3)
(4)
(5)
Lag
number
Dynamic
multipliers
Dynamic
multipliers
Dynamic
multipliers
Dynamic
multipliers
Dynamic
multipliers
0
–0.9
(1.3)
–1.1
(1.3)
–1.3
(1.7)
–0.2
(1.7)
–2.0
(1.5)
1
3.5
(1.6)
3.2
(1.6)
1.8
(1.6)
0.8
(1.5)
–
2
–1.3
(1.7)
–3.0
(1.6)
–2.2
(1.4)
–
–
3
0.2
(1.7)
1.5
(1.2)
–
–
–
4
–2.0
(1.5)
–
–
–
–
BIC
–234.4
–236.1
–238.5
–240.0
–241.8
(a) Which of these models is favored by the information criterion?
(b) How consistent are these estimates with the theory? Is this a fair test of the theory? Why or why not?
(c) Can you think of any improved data to test the theory?
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2) The Gallup Poll frequently surveys the electorate to quantify the public’s opinion of the president.
Since 1945, Gallup settled on the following wording of its presidential poll: “Do you approve or
disapprove of the way (name) is handling his job as president?” Gallup has not changed its presidential
question since then, and respondents can answer “approve,” “disapprove,” or “no opinion.”
You want to see how this approval rating is related to the Michigan index of consumer sentiment (ICS).
The monthly survey, conducted with a minimum sample of 500, asks people if they feel “better/worse off”
with regard to current and future conditions.
(a) To estimate dynamic causal effects, you collect quarterly data from 1962:I – 1998:II for the United
States. You allow a binary variable for each presidency to capture the intrinsic popularity of the
President. Furthermore, you eliminate observations that include a change in party for the presidency by
using a binary variable, which takes on the value of one during the first quarter of the year after the
election. Finally, a friendly political scientist provides you with (i) an “events” variable, (ii) a “Vietnam”
binary variable, and (iii) a “honeymoon” variable, which measures the effect of a higher popularity of a
president immediately following the election. (The coefficients of these variables will not be reported
here.)
Assuming that consumer sentiment is exogenous, you estimate the following two specifications (numbers
in parenthesis are heteroskedasticity– and autocorrelation–consistent standard errors):
t = 26.08 + 0.178 × ICSt + 0.232 × ICSt–1; R2= 0.667, SER = 7.00
(8.83) (0.120) (0.135)
t = 26.08 + 0.178 × ΔICSt + 0.411 + ICSt–1; R2 = 0.667, SER = 7.00
(8.17) (0.120 ) (0.089)
What is the difference between the two specifications? What is the advantage of estimating the second
equation, if any?
(b) Assuming that the errors follow an AR(1) process, you also estimate the following alternative:
t = –4.61 + 0.300 × ICSt – 0.070 × ICSt–1– 0.054 × ICSt–2; + 0.776 × Approvalt–1;
(5.84) (0.083) (0.099) (0.083) (0.057)
R2 = 0.868, SER = 4.45
How is this specification related to the previous ones? What implicit assumptions did you have to make
to allow for desirable properties of the OLS estimator?
(c) You finally estimate the approval equation using the quasi–difference specification and the GLS
estimator.
How is this equation related to the ones in (a) and (b)? What are the properties of the GLS estimator here,
under the assumption that ICS is strictly exogenous?
(d) Is it likely that the ICS is exogenous here? Strictly exogenous?
3) Consider the following distributed lag model is
serially uncorrelated, and X is strictly exogenous.
(a) How many parameters are there to be estimated between the two equations?
(b) Using the two equations of the model above, derive the ADL form of the model.
(c) There are five regressors in the ADL model, namely Yt–1, Xt, Xt–1, Xt–2 and the constant. Estimating
the ADL model linearly will give you five coefficients. Can you derive the parameters of the original two
equation model from these five estimates? Why or why not?
(d) What alternative method do you have to retrieve the parameters of the two equation model?
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4) A model that attracted quite a bit of interest in macroeconomics in the 1970s was the St. Louis model.
The underlying idea was to calculate fiscal and monetary impact and long run cumulative dynamic
multipliers, by relating output (growth) to government expenditure (growth) and money supply
(growth). The assumption was that both government expenditures and the money supply were
exogenous. Estimation of a St. Louis type model using quarterly data from 1960:I–1995:IV results in the
following output (HAC standard errors in parenthesis):
t = 0.018 + 0.006 × dmgrowtht + 0.235 × dmgrowtht–1 + 0.344 × dmgrowtht–2
(0.004) (0.079) (0.091) (0.087)
+ 0.385 × dmgrotht–3 + 0.425 × mgrowtht–4 + 0.170 × dggrowtht – 0.044dggrowtht–1
(0.097) (0.069) (0.049) (0.068)
– 0.003 × dggrowtht–2 – 0.079 × dggrowtht–3 + 0.018 × ggrowtht–4;
(0.040) (0.051) (0.027)
R2 = 0.346, SER=0.03
where ygrowth is quarterly growth of real GDP, mgrowth is quarterly growth of real money supply (M2),
and ggrowth is quarterly growth of real government expenditures. “d” in front of ggrowth and mgrowth
indicates a change in the variable.
(a) Assuming that money and government expenditures are exogenous, what do the coefficients
represent? Calculate the h–period cumulative dynamic multipliers from these. How can you test for the
statistical significance of the cumulative dynamic multipliers and the long–run cumulative dynamic
multiplier?
(b) Sketch the estimated dynamic and cumulative dynamic fiscal and monetary multipliers.
(c) For these coefficients to represent dynamic multipliers, the money supply and government
expenditures must be exogenous variables. Explain why this is unlikely to be the case. As a result, what
importance should you attach to the above results?