Exam 14: Introduction to Time Series Regression and Forecasting

An autoregression is a regression

To choose the number of lags in either an autoregression or in a time series regression model with multiple predictors, you can use any of the following test statistics with the Exception of the

(Requires Appendix Material) Define the difference operator $\Delta = ( 1 - L )$ , where $L$ is the lag operator, such that $L ^ { j } Y _ { t } = Y _ { t - j }$ . In general, $\Delta _ { j } ^ { i } = \left( 1 - L ^ { j } \right) ^ { i }$ , where $i$ and $j$ are typically omitted when they take the value of 1. Show the expressions in $Y$ only when applying the difference operator to the following expressions, and give the resulting expression an economic interpretation, assuming that you are working with quarterly data: (a) $\Delta _ { 4 } Y _ { t }$

(Requires Appendix material): Show that the AR(1) process $Y _ { t } = a _ { 1 } Y _ { t - 1 } + e _ { t } ; \left| a _ { 1 } \right| < 1$ be converted to a $\operatorname { MA } ( \infty )$ process.

The time interval between observations can be all of the following with the exception of data collected

The following two graphs give you a plot of the United States aggregate unemployment rate for the sample period 1962:I to 1999:IV, and the (log)level of real United States GDP for the sample period 1962:I to 1995:IV.You want test for stationarity in both cases.Indicate whether or not you should include a time trend in your Augmented Dickey-Fuller test and why. United States Unemployment Rate United States Real GDP (in logarithms)

Consider the following model $Y _ { t } = \alpha _ { 0 } + \alpha _ { 1 } X _ { t } ^ { e } + u _ { t }$ t where the superscript "e" indicates expected values.This may represent an example where consumption depended on expected, or "permanent," income.Furthermore, let expected income be formed as follows: $X _ { t } ^ { e } = X _ { t - 1 } ^ { e } + \lambda \left( X _ { t - 1 } - X _ { t - 1 } ^ { e } \right) ; 0 < \lambda < 1$ 1 This particular type of expectation formation is called the "adaptive expectations hypothesis." (a) In the above expectation formation hypothesis, expectations are formed at the beginning of the period, say the $1 ^ { \text {st } }$ of January if you had annual data. Give an intuitive explanation for this process.

Departures from stationarity

(Requires Appendix material) The long-run, stationary state solution of an $\operatorname { AD } ( p , q )$ model, which can be written as $A ( L ) Y _ { t } = \beta _ { 0 } + c ( L ) X _ { t - 1 } + u _ { t }$ , where $a _ { 0 } = 1$ , and $a _ { j } = - \beta _ { j }$ , $c _ { j } = \delta _ { j }$ , can be found by setting $L = 1$ in the two lag polynomials. Explain. Derive the long-run solution for the estimated $\operatorname { ADL } ( 4,4 )$ of the change in the inflation rate on unemployment: =1.32-.36\Delta Inf -0.34\Delta+.07\Delta Inf -.03\Delta Inf -2.68+3.43-1.04+.07 Assume that the inflation rate is constant in the long-run and calculate the resulting unemployment rate.What does the solution represent? Is it reasonable to assume that this long-run solution is constant over the estimation period 1962-1999? If not, how could you detect the instability?

Pseudo out of sample forecasting can be used for the following reasons with the exception of

You have decided to use the Dickey Fuller (DF)test on the United States aggregate unemployment rate (sample period 1962:I - 1995:IV).As a result, you estimate the following AR(1)model = 0.114-0.024,=0.0118, SER =0.3417 (0.121)(0.019) You recall that your textbook mentioned that this form of the AR(1)is convenient because it allows for you to test for the presence of a unit root by using the t- statistic of the slope.Being adventurous, you decide to estimate the original form of the AR(1) instead, which results in the following output = 0.114-0.024,=0.0118, SER =0.3417 (0.121)(0.019) You are surprised to find the constant, the standard errors of the two coefficients, and the SER unchanged, while the regression $\mathrm { R } ^ { 2 }$ increased substantially. Explain this increase in the regression $\mathrm { R } ^ { 2 }$ . Why should you have been able to predict the change in the slope coefficient and the constancy of the standard errors of the two coefficients and the SER?

The first difference of the logarithm of $Y _ { t }$ equals

The random walk model is an example of a

You set out to forecast the unemployment rate in the United States (UrateUS), using quarterly data from 1960, first quarter, to 1999, fourth quarter. (a)The following table presents the first four autocorrelations for the United States aggregate unemployment rate and its change for the time period 1960 (first quarter)to 1999 (fourth quarter).Explain briefly what these two autocorrelations measure. First Four Autocorrelations of the U.S. Unemployment Rate and its Change, 1960:I - 1999 IV Unemployment Rate Change of Unemployment Rate 1 0.97 0.62 2 0.92 0.32 3 0.83 0.12 4 0.75 -0.07

The textbook displayed the accompanying four economic time series with "markedly different patterns." For each indicate what you think the sample autocorrelations of the level (Y)and change ( ΔY )will be and explain your reasoning. (a)

$\text { Consider the standard AR(1) } Y _ { t } = \beta _ { 0 } + \beta _ { 1 } Y _ { t - 1 } + u _ { t } \text {, where the usual assumptions hold. }$ (a) Show that $y _ { t } = \beta _ { 1 } y _ { t - 1 } + u _ { t }$ , where $y _ { t }$ is $Y _ { t }$ with the mean removed, i.e., $y _ { t } = Y _ { t } - E \left( Y _ { t } \right)$ . Show that $E \left( y _ { t } \right) = 0$ .

The AR(p) model

Problems caused by stochastic trends include all of the following with the exception of

You collect monthly data on the money supply (M2) for the United States from 1962:12002:4 to forecast future money supply behavior. where $L M 2$ and $D L M 2$ are the log level and growth rate of M2. (a)Using quarterly data, when analyzing inflation and unemployment in the United States, the textbook converted log levels of variables into growth rates by differencing the log levels, and then multiplying these by 400.Given that you have monthly data, how would you proceed here?

The Augmented Dickey Fuller (ADF)t-statistic