Introduction to Econometrics, 3e (Stock)
Chapter 16 Additional Topics in Time Series Regression
16.1 Multiple Choice
1) A vector autoregression
A) is the ADL model with an AR process in the error term.
B) is the same as a univariate autoregression.
C) is a set of k time series regressions, in which the regressors are lagged values of all k series.
D) involves errors that are autocorrelated but can be written in vector format.
2) A multiperiod regression forecast h periods into the future based on an AR(p) is computed
A) the same way as the iterated AR forecast.
B) by estimating the multiperiod regression Yt = δ0 + δ1Yt–h + … + δpYt–p–h+1 + ut, then using the
estimated coefficients to compute the forecast h periods in advance.
C) by estimating the multiperiod regression Yt = δ0 + δ1Yt–h + ut , then using the estimate coefficients to
compute the forecast h period in advance.
D) by first computing the one–period ahead forecast, next using that to compute the two–period ahead
forecast, and so forth.
3) Multiperiod forecasting with multiple predictors
A) is the same as the iterated AR forecast method.
B) can use the iterated VAR forecast method.
C) will yield superior results when using the multiperiod regression forecast h periods into the future
based on p lags of each Yt , rather than the iterated VAR forecast method.
D) will always yield superior results using the iterated VAR since it takes all equations into account.
4) If Yt is I(2), then
A) Δ2Yt is stationary.
B) Yt has a unit autoregressive root.
C) ΔYt is stationary.
D) Yt is stationary.
5) The following is not a consequence of Xt and Yt being cointegrated:
A) if Xt and Yt are both I(1), then for some θ, Yt – θ Xt is I(0).
B) Xt and Yt have the same stochastic trend.
C) in the expression Yt – θ Xt , θ is called the cointegrating coefficient.
D) if Xt and Yt are cointegrated then integrating one of the variables gives you the same result as
integrating the other.
6) One advantage of forecasts based on a VAR rather than separately forecasting the variables involved is
A) that VAR forecasts are easier to calculate.
B) you typically have knowledge of future values of at least one of the variables involved.
C) it can help to make the forecasts mutually consistent.
D) that VAR involves panel data.
7) The coefficients of the VAR are estimated by
A) using a simultaneous estimation method such as TSLS.
B) maximum likelihood.
C) panel methods.
D) estimating each of the equations by OLS.
8) Under the VAR assumptions, the OLS estimators are
A) consistent and have a joint normal distribution even in small samples.
B) BLUE.
C) consistent and have a joint normal distribution in large samples.
D) unbiased.
9) A VAR allows you to test joint hypothesis that involve restrictions across multiple equations by
A) computing a z–statistic.
B) computing the BIC but not the AIC.
C) using a stability test.
D) computing an F–statistic.
10) A VAR with five variables, 4 lags and constant terms for each equation will have a total of
A) 21 coefficients.
B) 100 coefficients.
C) 105 coefficients.
D) 84 coefficients.
11) You can determine the lag lengths in a VAR
A) by using confidence intervals.
B) by using critical values from the standard normal table.
C) by using either F–tests or information criteria.
D) with the help from economic theory and institutional knowledge.
12) The biggest conceptual difference between using VARs for forecasting and using them for structural
modeling is that
A) you need to use the Granger causality test for structural modeling.
B) structural modeling requires very specific assumptions derived from economic theory and institutional
knowledge of what is exogenous and what is not.
C) you can no longer use the information criteria to decide on the lag length.
D) structural modeling only allows a maximum of three equations in the VAR.
13) The error term in a multiperiod regression
A) is serially correlated.
B) causes OLS to be inconsistent.
C) is serially correlated, but less so the longer the forecast horizon.
D) is serially uncorrelated.
14) Δ2Yt
A) = ΔYt – ΔYt–1.
B) = – .
C) = ΔYt –ΔYt–2.
D) = Yt – Yt–2.
15) The order of integration
A) can never be zero.
B) is the number of times that the series needs to be differenced for it to be stationary.
C) is the value of φ1 in the quasi difference(ΔYt – φ1Yt–1).
D) depends on the number of lags in the VAR specification.
16) To test the null hypothesis of a unit root, the ADF test
A) has higher power than the so–called DF–GLS test.
B) uses complicated interative techniques.
C) cannot be calculated if the variable is integrated of order two or higher.
D) uses a t–statistic and a special critical value.
17) Unit root tests
A) use the standard normal distribution since they are based on the t–statistic.
B) cannot use the standard normal distribution for statistical inference. As a result the ADF statistic has its
own special table of critical values.
C) can use the standard normal distribution only when testing that the level variable is stationary, but not
the difference variable.
D) can use the standard normal distribution but only if HAC standard errors were computed.
18) In a VECM,
A) past values of Yt – θ Xt help to predict future values of ΔYt and/or ΔXt.
B) errors are corrected for serial correlation using the Cochrane–Orcutt method.
C) current values of Yt – θ Xt help to predict future values of ΔYt and/or ΔXt.
D) VAR techniques, such as information criteria, no longer apply.
19) The following is not an appropriate way to tell whether two variables are cointegrated:
A) see if the two variables are integrated of the same order.
B) graph the series and see whether they appear to have a common stochastic trend.
C) perform statistical tests for cointegration.
D) use expert knowledge and economic theory.
20) If Xt and Yt are cointegrated, then the OLS estimator of the coefficient in the cointegrating regression
is
A) BLUE.
B) unbiased when using HAC standard errors.
C) unbiased even in small samples.
D) consistent.
21) Assume that you have used the OLS estimator in the cointegrating regression and test the residual for
a unit root using an ADF test. The resulting ADF test statistic has a
A) normal distribution in large samples.
B) non–normal distribution which requires ADF critical values for inference.
C) non–normal distribution which requires EG–ADF critical values for inference.
D) normal distribution when HAC standard errors are used.
22) The DOLS estimator has the following property if Xt and Yt are cointegrated:
A) it is BLUE even in small samples.
B) it is efficient in large samples.
C) it has a standard normal distribution when homoskedasticity–only standard errors are used.
D) it has a non–normal distribution in large samples when HAC standard errors are used.
23) Volatility clustering
A) is evident in most cross–sections.
B) implies that a series is serially correlated.
C) can mostly be found in studies of the labor market.
D) is evident in many financial time series.
24) Using the ADL(1,1) regression Yt = β0 + β1Yt–1 + Xt–1 + ut, the ARCH model for the regression
error assumes that ut is normally distributed with mean zero and variance , where
A) = α0 + α1 + α2 + … + αp.
B) = + … + + φ1+ … + φq.
C) = φ1+ … + φq.
D) = α0 + α1+ … + αp+ φ1+ … + φq.
25) ARCH and GARCH models are estimated using the
A) OLS estimation method.
B) the method of maximum likelihood.
C) DOLS estimation method.
D) VAR specification.
26) A VAR with k time series variables consists of
A) k equations, one for each of the variables, where the regressors in all equations are lagged values of all
the variables
B) a single equation, where the regressors are lagged values of all the variables
C) k equations, one for each of the variables, where the regressors in all equations are never more than
one lag of all the variables
D) k equations, one for each of the variables, where the regressors in all equations are current values of all
the variables
27) The BIC for the VAR is
A) BIC(p) = ln[det ( u)] + k(kp+1)
B) BIC(p) = ln[det ( u)] + k(p+1)
C) BIC(p) = ln[det ( u)] + k(kp+1)
D) BIC(p) = ln[SSR(p)] + k(p+1)
28) The lag length in a VAR using the BIC proceeds as follows: Among a set of candidate values of p, the
estimated lag length xxxis the value of p
A) For which the BIC exceeds the AIC
B) That maximizes BIC(p)
C) Cannot be determined here since a VAR is a system of equations, not a single one
D) That minimizes BIC(p)
29) The dynamic OLS (DOLS) estimator of the cointegrating coefficient, if Yt and Xt are cointegrated,
A) is efficient in large samples
B) statistical inference about the cointegrating coefficient is valid
C) the t–statistic constructed using the DOLS estimator with HAC standard errors has a standard normal
distribution in large samples
D) all of the above
30) The EG–ADF test
A) is the similar to the DF–GLS test
B) is a test for cointegration
C) has as a limitation that it can only test if two variables, but not more than two, are cointegrated
D) uses the ADF in the second step of its procedure
16.2 Essays and Longer Questions
1) “Heteroskedasticity typically occurs in cross–sections, while serial correlation is typically observed in
time–series data.” Discuss and critically evaluate this statement.
2) Some macroeconomic theories suggest that there is a short–run relationship between the inflation rate
and the unemployment rate. How would you go about forecasting these two variables? Suggest various
alternatives and discuss their advantages and disadvantages.
3) Think of at least five examples from economics where theory suggests that the variables involved are
cointegrated. For one of these cases, explain how you would test for cointegration between the variables
involved and how you could use this information to improve forecasting.
4) What role does the concept of cointegration and the order of integration play in modeling the
relationship between variables? Explain how tests of cointegration work.
5) Carefully explain the difference between forecasting variables separately versus forecasting a vector of
time series variables. Mention how you choose optimal lag lengths in each case. Part of your essay should
deal with multiperiod forecasts and different methods that can be used in that situation. Finally address
the difference between VARS and VECM.
6) You have collected quarterly data for the unemployment rate (Unemp) in the United States, using a
sample period from 1962:I (first quarter) to 2009:IV (the data is collected at a monthly frequency, but you
have taken quarterly averages).
a. Does economic theory suggest that the unemployment rate should be stationary?
b. Testing the unemployment rate for stationarity, you run the following regression (where the lag
length was determined using the BIC; using the AIC instead does not change the outcome of the test,
even though it chooses 9 lags of the LHS variable):
t = 0.217 – 0.035 Unempt–1 + 0.689 ΔUnempt–1
(0.01) 0.0012) (0.054)
Use the ADF statistic with an intercept only to test for stationarity. What is your decision?
c. The standard errors reported above were homoskedasticity–only standard errors. Do you think you
could potentially improve on inference by allowing for HAC standard errors?
d. An alternative test for a unit root, the DF–GLS, produces a test statistic of –2.75. Find the critical value
and decide whether or not to reject the null hypothesis. If the decision is different from (c), is there any
reason why you might prefer the DF–GLS test over the ADF test?
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16.3 Mathematical and Graphical Problems
1) Consider the GARCH(1,1) model = α0 + α1+ φ1. Show that this model can be rewritten
as = + α1(+ φ1 + + + …). (Hint: use the GARCH(1,1) model but
specify it for ; substitute this expression into the original specification, and so on.) Explain
intuitively the meaning of the resulting formulation.
2) You have collected quarterly data on inflation and unemployment rates for Canada from 1961:III to
1995:IV to estimate a VAR(4) model of the change in the rate of inflation and the unemployment rate. The
results are
t = 1.02 – .54 ΔInft–1 – .46 ΔInft–2 – .32 ΔInft–2 – .01 ΔInft–4
.09) (.09) (.09) (.08) (.44)
–.76 Unempt–1 + .20 Unempt–2 – .16 Unempt–3 + .59 Unempt–4
(.43) (.76) (.76) (.44)
= .26.
t = 0.18 – .003 ΔInft–1 – .016 ΔInft–2 – .018 ΔInft–3 – .010 ΔInft–4
(.10) (.016) (.018) (.017) (.016)
+ 1.47 Unempt–1 – .46 Unempt–2 – .08 Unempt–3 + .05 Unempt–4
(.08) (.14) (.14) (.08)
= .980.
(a) Explain how you would use the above regressions to conduct one period ahead forecasts.
(b) Should you test for cointegration between the change in the inflation rate and the unemployment rate
and, in the case of finding cointegration here, respecify the above model as a VECM?
(c) The Granger causality test yields the following F–statistics: 3.75 for the test that the coefficients on
lagged unemployment rate in the change of inflation equation are all zero; and 0.36 for the test that the
coefficients on lagged changes in the inflation rate are all zero. Based on these results, does
unemployment Granger–cause inflation? Does inflation Granger–cause unemployment?