Exam 16: Additional Topics in Time Series Regression

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.

Under the VAR assumptions, the OLS estimators are

To test the null hypothesis of a unit root, the ADF test

Volatility clustering

There has been much talk recently about the convergence of inflation rates between many of the OECD economies.You want to see if there is evidence of this closer to home by checking whether or not Canada's inflation rate and the United States' inflation rate are cointegrated. (a)You begin your numerical analysis by testing for a stochastic trend in the variables, using an Augmented Dickey-Fuller test.The t-statistic for the coefficient of interest is as follows: Variable with lag of 1 InfCan \Delta InfCan InfUS \Delta InfUS t -statistic -1.93 -6.38 -2.37 -5.63 where InfCan is the Canadian inflation rate, and InfUS is the United States inflation rate. The estimated equation included an intercept.For each case make a decision about the stationarity of the variables based on the critical value of the Augmented Dickey-Fuller test statistic.

Unit root tests

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

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

ARCH and GARCH models are estimated using the

You can determine the lag lengths in a VAR

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

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.

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.

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 (a)Explain how you would use the above regressions to conduct one period ahead forecasts.

You have collected time series for various macroeconomic variables to test if there is a single cointegrating relationship among multiple variables.Formulate the null hypothesis and compare the EG-ADF statistic to its critical value. (a)Canadian unemployment rate, Canadian Inflation Rate, United States unemployment rate, United States inflation rate; t = (-3.374).

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 $\operatorname { ADL } ( 3,1 )$ formulation of the Phillips curve using a $\mathrm { GARCH } ( 1,1 )$ specification. The results are as follows: = 1.17-.56\Delta Inf -.47\Delta Inf -.31\Delta-.13 (.48)(.08)(.10)(.09)(.06) = .86+.27+.53. (.40)(.11)(.15) (a) Test the two coefficients for $u _ { t - 1 } ^ { 2 }$ and $\sigma _ { t - 1 } ^ { 2 }$ in the GARCH model individually for statistical significance.

The coefficients of the VAR are estimated by

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?