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1、Introduction to Econometrics, 3e (Stock)Chapter 16 Additional Topics in Time Series Regression16.1 Multiple Choice1) A vector autoregressionA) 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 re
2、gressors are lagged values of all k series.D) involves errors that are autocorrelated but can be written in vector format.Answer: C2) A multiperiod regression forecast h periods into the future based on an AR(p) is computedA) the same way as the iterated AR forecast.B) by estimating the multiperiod
3、regression Yf = do + 小+ + pt-p-h+1 + then using the estimated coefficients to compute the forecast h periods in advance.C) by estimating the multiperiod regression Yf =+ di Yf_/t + U, then using the estimate coefficients tocompute the forecast h period in advance.D) by first computing the one-period
4、 ahead forecast, next using that to compute the two-period ahead forecast, and so forth.Answer: B3) Multiperiod forecasting with multiple predictorsA) 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
5、regression forecast h periods into the future based on p lags of each Yf, rather than the iterated VAR forecast method.D) will always yield superior results using the iterated VAR since it takes all equations into account.Answer: B4) If Yt is 1(2), thenA) A2Yf is stationary.B) Yf has a unit autoregr
6、essive root.C) AYf is stationary.D) Yf is stationary.Answer: A5) The following is not a consequence of Xf and Yf being cointegrated:A) if Xf and Yf are both /(I), then for some 0, Yf - 0 Xf is 1(0).B) Xf and Yf have the same stochastic trend.C) in the expression Yf- 0 Xf, 0 is called the cointegrati
7、ng coe仔icient.D) if Xf and Yf are cointegrated then integrating one of the variables gives you the same result as integrating the other.Answer: D5) Carefully explain the difference between forecasting variables separately versus forecasting a vector of time series variables. Mention how you choose o
8、ptimal 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.Answer: When variables are forecasted separately, then single equations of the AR(p) type are typi
9、cally involved. If economic theory and/or institutional knowledge suggest that additional predictors should be included, then forecasts can be potentially improved by estimating an ADL(p,q) model. For one period ahead forecasts, these are identical to forecasts based on systems of equations. Lag len
10、gths will be chosen using the BIC or the AIC criterium.There are three important reasons why VARs may be preferable for forecasting. One results from the forecasting horizon. If forecasts are to be made two or more periods ahead, then if future values of the additional predictors are to be used, the
11、se have to be forecasted themselves. This can be avoided by choosing the multiperiod regression method. Here, in the case of an h period forecast, multiperiod regressions are estimated where all predictors are lagged h periods or more. Second, using VAR forecasting methods will make the forecasts fo
12、r the variables involved mutually consistent. This is the result of using the iterated VAR forecasts whereby the forecasted values are subsequently used to forecast further ahead. Finally VAR models allow for restrictions across equations to be tested.Multiperiod regression methods in general may be
13、 preferable over iterated forecasts if the AR(p), ADL(p,q) or VAR models are incorrectly specified. In practice, the difference in forecasts tends to be very small between the multiperiod regression and iterated forecast methods.VAR models can be enhanced by incorporating long-run information in the
14、 form of error correction terms. If some of the variables in the VAR model have a common stochastic trend, then this can be used to improve the forecasts by including the error correction term, thereby turning the VAR model into a VECM.6) You have collected quarterly data for the unemployment rate (
15、Unemp) in the United States, using a sample period from 1962:1 (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 station
16、arity, 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):= 0.217 - 0.035 Unempt-i + 0.689 AUnempt-i(0.01)0.0012)(0.054)Use the ADF statistic with an in
17、tercept 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 te
18、st 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?Answer:a. In macroeconomics or labor economics, you have learned about the natural
19、 rate of unemployment, or the Non-Accelerating Inflation Rate of Unemployment (NAIRU). The idea here is that unemployment rates may deviate from this equilibrium unemployment rate, but that, following a shock, the unemployment rate will revert towards this equilibrium. Hence you might expect the dif
20、ference between the unemployment rate and the NAIRU, referred by some as the cyclical unemployment rate, to be stationary. Unfortunately the equilibrium unemployment rate is not a constant over time and may be affected by demographics, the price of search (unemployment insurance benefits), and other
21、 variables. If the NAIRU is not a constant over time, then the unemployment rate itself may not be stationary. Furthermore, there is also the idea of hysteresis, which allows for the unemployment rate to move to a new equilibrium rate once a shock hits the economy. The bottom line is that while ther
22、e is some guidance from economic theory, it is an empirical question whether or not the unemployment rate is stationary.b. The t-statistic for the ADF test is -2.84. The critical value at the 5% level is -2.86. Hence you can reject the null hypothesis of a unit root for the unemployment rate at the
23、10% level, but (just) fail to reject the null hypothesis at the 5% level. Most economist treat the unemployment rate as stationary.c. The ADF statistic is computed using non-robust standard errors. It turns out that under the null hypothesis of a unit root, the homoskedasticty-only standard errors g
24、enerate a t-statistic that is robust to heteroskedasticityd. The critical value for the DF-GLS test is -2.58 at the 1% level. Hence you can reject the null hypothesis of a unit root using this test. The DF-GLS has a higher power when compared to the ADF test, and hence should be preferred.16.3 Mathe
25、matical and Graphical Problems2221) Consider the GARCH(1,1) model o 1=a。+ 戊1 ,_ + 01 0 /_. Show that this model can be rewritten2。22, 2 2. 3 9as er+ 戊1( u /_ + 01 卜2 + i f-3 + i “ f-4 + )(Hint: use the GARCH(1,1) model2but specify it for。substitute this expression into the original specification, an
26、d so on.) Explain intuitively the meaning of the resulting formulation.222222Answer: o ,=碰 + a /_ + 1 0 卜二碰 + a 卜+ 1(戊0 + 戊1 f_2+ 轲 0 f-2)222 2二 0(1 + 阿)+ 州(bi + 01 f_2)+ 01。t-22222 23 2=仪0(1 + 01 + 0 ) + 1( /_ + Ql 卜2 + $ 1 f-3)+ 0 t-3. Continuing with the substitutions231infinitely and noting that
27、 the sum of the geometric series is 1+ Qi + 0 +0 + =you finally2 ao 222 23 2arrive at (7+ al( j2 + S u t-2 + 0 ,3+ 0 ,4+ -)- This expression states thatthe variances depend on a weighted average of past squared residuals, where the distant past receives a smaller weight than more recently observed s
28、quared residuals.2) You have collected quarterly data on inflation and unemployment rates for Canada from 1961:111 to 1995:IV to estimate a VAR(4) model of the change in the rate of inflation and the unemployment rate. The results areXlnft = 1.02 - .54 Mnft-i - .46 AIn/f-2 - .32 Aln/f-2 - .。1 呵t-4.0
29、9) (.09)(.09)(.08)(.44)-.76 Unemp.i + .20 Unemp.2 - -16 Unempt_3 + .59 Unemp(.43)(.76)(.76)(.44)铲=.26.Unent= 0.18 - .003 /班-1 - .016 AInft-2 - .。18 A/n/f-3 - .010 Aht/f-4(.10) (.016)(.018)(.017)(.016)+ 1.47 Unempi-i - .46 Unempt-2 - .08 Unempt-3 + 5 Unempt-4(.08)(.14)(.14)(.08)R2 = .980.(a) Explain
30、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 yi
31、elds 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 coe仔icients on lagged changes in the inflation rate are all zero. Based on these results, does unemployment Granger-ca
32、use inflation? Does inflation Granger-cause unemployment?Answer:(a) One period ahead forecasts are the same as for the ADL(4,4) models of the inflation rate and unemployment rate. For example, forecasting the change in the inflation rate for 1996:1 requires use of the actual values for unemployment
33、and change in inflation rates through 1995:IV. The unemployment rate for 1996:1 is forecasted in the same way using the second regression.(b) Most economic theories suggest that there is no long-run relationship between the inflation rate and the unemployment rate, or, stated differently, that the l
34、ong-run Phillips curve is vertical. Hence economic theory does not suggest testing for cointegration or using the error correction term in a VECM model.(c) The critical value for the14,8 statistic is 3.32 at the 1% significance level, and 1.94 at the 10% significance level. Based on the calculated F
35、-statistics above you can reject the null hypothesis that lagged unemployment rates do not Granger-cause the inflation rate, but you cannot reject the null hypothesis that lagged inflation does not Granger-cause the unemployment rate.3) Purchasing power parity (PPP), postulates that the exchange rat
36、e between two countries equals the pfratio of the respective price indexes or ExchRate = (where ExchRate is the foreign exchange ratebetween the two countries, and P represents the price index, with/indicating the foreign country). The long-run version of PPP implies that that the exchange rate and
37、the price ratio share a common trend.(a) You collect monthly foreign exchange rate data from 1974:1 to 2002:4 for the U.S./U.K. exchange rate ($/) and you collect data on the Consumer Price Index for both countries. Explain how you would used the Engle-Granger test statistic to investigate the long-
38、run PPP hypothesis.(b) One of your peers explains that there may be an easier way to test for the validity of PPP. She suggests to simply test whether or not the real exchange rate, or competitiveness, is stationary. (The pfreal exchange rate is given by ExchRate x ) Is she correct? Explain. How wou
39、ld you implement hersuggestion? Which alternative test-statistic is available?Answer:(c) Using the Engle-Granger two step procedure, the (log of) the exchange rate will be regressed on the relative price ratio (log difference of the two prices). The residuals from this regression will then be subjec
40、ted to a Dickey-Fuller t-test with an intercept but no time trend. This is the EG-ADF procedure. However, the OLS estimator of the coefficient in this regression is only consistent if the two variables are cointegrated. Furthermore, inference can be misleading since the OLS estimator does not have a
41、 normal distribution. If a test is performed on whether the coefficient of the price ratio is unity, then the DOLS estimator should be used with HAC standard errors.(d) If PPP holds, then the exchange rate and the relative price ratio will have a cointegrating coefficient of 0 = 1. First the real ex
42、change rate should be plotted to inspect visually whether or not the two variables are cointegrated. To test this more formally, the real exchange rate should be tested for containing a unit root, using the ADF statistic. If the null hypothesis is rejected, then this would suggest that PPP holds in
43、the long-run. Since the ADF test is not the most powerful test, the DF-GLS test can be used as an alternative.A 22o；=.86 + .27；i+ .53(J2t-r4) You have collected quarterly Canadian data on the unemployment and the inflation rate from 1962:1 to 2001 :IV. You want to re-estimate the ADL(3,1) formulatio
44、n of the Phillips curve using a GARCH(1,1) specification. The results are as follows:Mnft = 1.17 - .56 Alnft.i - .47 A/n/f-2 - .31 Xnft-3 - .13 Unempt-1(.06)(.48) (.08)(.10)(.09)(40) (.11)(.15)22(a) Test the two coefficients for and 0 in the GARCH model individually for statistical significance.(b)
45、Estimating the same equation by OLS results in/n/ = 1.19 - .51 AInff-i - .47 Mnft-2 - .28 AInft-3 - .16Uempt-i(07)(.54) (.10)(.11)(.08)Briefly compare the estimates. Which of the two methods do you prefer?(c) Given your results from the test in (a), what can you say about the variance of the error t
46、erms in the Phillips Curve for Canada?(d) The following figure plots the residuals along with bands of plus or minus one predicted standard deviation (that is, 土前 based on the GARCH(1,1) model.RESIDUALS STDPLUS STDMINUSDescribe what you see.Answer:(a) The two f-statistics are 2.46 and 3.53 respectiv
47、ely. Since they are normally distributed in large samples you can use the standard normal distribution for significance testing and the construction of confidence intervals. The first is coefficient statistically significant at the 5% level, while the second is statistically significant at the 1% le
48、vel.(b) These are two estimation methods, OLS and Maximum Likelihood. The GARCH(1,1) model produces very similar estimates for the lagged inflation and unemployment rates. The deference stems from the fact that the two GARCH coefficients are (significantly) different from zero. Since they are statistically significant, GARCH is the preferred model since it does not constrain the coefficients to zero.(c) The tests in (a) suggest that the errors are not homoskedastic but conditionally heteroskedastic.(d)