Exam 16: Additional Topics in Time Series Regression

For the United States,there is somewhat conflicting evidence whether or not the inflation rate has a unit autoregressive root.For example,for the sample period 1962:I to 1999:IV using the ADF statistic,you cannot reject at the 5% significance level that inflation contains a stochastic trend.However the null hypothesis can be rejected at the 10% significance level.The DF-GLS test rejects the null hypothesis at the five percent level.This result turns out to be sensitive to the number of lags chosen and the sample period. (a)Somewhat intrigued by these findings,you decide to repeat the exercise using Canadian data.Letting the AIC choose the lag length of the ADF regression,which turns out to be three,the ADF statistic is (-1.91).What is your decision regarding the null hypothesis? (b)You also calculate the DF-GLS statistic,which turns out to be (-1.23).Can you reject the null hypothesis in this case? (c)Is it possible for the two test statistics to yield different answers and if so,why?

Your textbook so far considered variables for cointegration that are integrated of the same order.For example,the log of consumption and personal disposable income might both be I(1)variables,and the error correction term would be I(0),if consumption and personal disposable income were cointegrated. (a)Do you think that it makes sense to test for cointegration between two variables if they are integrated of different orders? Explain. (b)Would your answer change if you have three variables,two of which are I(1)while the third is I(0)? Can you think of an example in this case?

If Yt is I(2),then

One advantage of forecasts based on a VAR rather than separately forecasting the variables involved is

Economic theory suggests that the law of one price holds.Applying this concept to foreign and domestic goods implies that goods will sell for the same price across countries.The consumer price index is the price for a basket of goods,and is calculated for countries as a whole.Hence in the absence of barriers to trade,and large transportation costs (and the fact that not all goods are traded)you should observe Purchasing Power Parity (PPP)between two countries,or ExchRate × P = Pf,where ExchRate is the foreign exchange rate between the two countries,and P represents the price index,with f indicating the foreign country.Dividing both sides of the equation by the domestic price level then gives you the standard formulation for PPP: ExchRate = .If PPP holds in the long run,then the exchange rate and the price ratio should share a common trend.Since it is a long-run concept,cointegration provides an interesting way to test for it. a.Using monthly data for the U.S./U.K.exchange rate ($/₤)and the respective price indexes,you estimate the following regression: t = 0.44 + 0.69 (lnPUS - lnPUK) Collecting the residuals from this regression and using an ADF test for cointegration,you find a t-statistic of -2.71.Can you reject the null-hypothesis of no cointegration? What is the critical value? b.Was it good econometric practice to test for cointegration right away? What else should you have done before proceeding with the EG-ADF test?

You have collected quarterly data for real GDP (Y)for the United States for the period 1962:I (first quarter)to 2009:IV. a.Testing the log of GDP for stationarity,you run the following regression (where the lag length was determined using the AIC): t = 0.03 - 0.0024 lnYt-1 + 0.253 ΔlnYt-1 + 0.167 ΔlnYt-2 (0.03)(0.0014)(0.072)(0.072) t = 1962:I - 2009:IV,R2 = 0.16,SER = 0.008 Use the ADF statistic with an intercept only to test for stationarity.What is your decision? b.You have decided to test the growth rate of real GDP for stationarity for the same sample period.The regression is as follows: t = 0.0041 - 0.543 ΔlnYt-1 - 0.186 Δ2lnYt-1 (0.0009)(0.082)(0.071) t = 1962:I - 2009:IV,R2 = 0.16,SER = 0.008 Use the ADF statistic with an intercept only to test for stationarity.What is your decision? c.Using the orders of integration terminology,what order of integration is the log level of real GDP? The growth rate? d.Given that the SER hardly changed in the second equation,why is the regression R2 larger?

ARCH and GARCH models are estimated using the

"Heteroskedasticity typically occurs in cross-sections,while serial correlation is typically observed in time-series data." Discuss and critically evaluate this statement.

Under the VAR assumptions,the OLS estimators are

A multiperiod regression forecast h periods into the future based on an AR(p)is computed

The error term in a multiperiod regression

A VAR allows you to test joint hypothesis that involve restrictions across multiple equations by

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.

The following is not a consequence of Xt and Yt being cointegrated:

The order of integration

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

You have collected quarterly Canadian data on the unemployment and the inflation rate from 1962:I to 2001:IV.You want to re-estimate the ADL(3,1)formulation of the Phillips curve using a GARCH(1,1)specification.The results are as follows: t = 1.17 - .56 ΔInft-1 - .47 ΔInft-2 - .31 ΔInft-3 - .13 Unempt-1 (.48)(.08)(.10)(.09)(.06) = .86 + .27 + .53 . (.40)(.11)(.15) (a)Test the two coefficients for and in the GARCH model individually for statistical significance. (b)Estimating the same equation by OLS results in t = 1.19 - .51 ΔInft-1 - .47 ΔInft-2 - .28 ΔInft-3 - .16Unempt-1 (.54)(.10)(.11)(.08)(.07) 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 terms 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,± )based on the GARCH(1,1)model. Describe what you see.

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.

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.

A VAR with five variables,4 lags and constant terms for each equation will have a total of