Univariate, Nonstationary Processes

Transcription

1 Univariate, Nonstationary Processes Jamie Monogan University of Georgia March 20, 2018 Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

2 Objectives By the end of this meeting, participants should be able to: Distinguish among varieties of nonstationarity. Run an augmented Dickey-Fuller test. Run a Kwiatkowski, Perron, Schmidt, and Shinn (KPSS) test. Use ARIMA processes to model integrated data. Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

3 The Nature of Nonstationarity Flavors of Nonstationarity All of our models so far have been for stationary series: A series is stationary if the mean and variance are constant over time, and the correlation between two observations depends only on the number of lags between them. When is this false? Two possibilities: 1 When a series is trending. 2 When a series contains a unit root. Figure source: Helmut Thome (2014) Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

4 The Nature of Nonstationarity Intuition About Stationarity Assume that y is a stationary series. That is, it tends to equilibrate to a fixed mean when disturbed. Now three cases for a particular y t 1 y t above equilibrium 2 y t at equilibrium 3 y t below equilibrium What do we expect of the change at t+1, y t+1? Thus we can write y t+1 = βy t + ɛ t. If y is stationary, then our estimated β will always be negative. (Think: How will y respond in each of the three cases?) So a negative β in a setup like this is evidence for stationarity as opposed to a unit root. What would we expect for a nonstationary series? Trend: A non-zero α term here: y t = α + ɛ t. (Because y t = η + αt + ν t.) Unit root: A zero β term here: y t = βy t + ɛ t. (Because y t = y t 1 + ɛ t.) Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

5 The Nature of Nonstationarity Running Example: Macropartisanship Macropartisanship Time Series Level Autocorrelation Function Does theory weigh in on the stationarity of this series? Or not? Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

6 Tests for Unit Roots Unit Root Testing For the AR(1)-like setup: y t = β 0 + β 1 y t 1 + u t It has a unit root if β 1 = 1.0 (exactly). Unit root implies that y is I(1). So we want to test for the null β 1 = 1.0 But if we simply estimate the equation, regression theory tells us that ˆβ 1 will be biased downwards. Hence, we have to get creative in how to test this. Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

7 Tests for Unit Roots Dickey-Fuller The Dickey-Fuller Solution y t = β 0 + γy t 1 + e t, for the null hypothesis that γ = 1.0. Subtract y t 1 from both sides, giving: y t - y t 1 = β 0 + (γ - 1)y t 1 + e t, and set β 1 = γ 1.0. Estimate: y t = β 0 + β 1 y t 1 + e t That is, predict the change in y at t from the level at t-1. Then the null becomes β 1 = 0.0 Now β 1 <0.0 indicates equilibration to the mean (stationarity). Negative and significant β 1 means that we do not have a unit root. Let us be clear: The null hypothesis is nonstationary. A significant result means we have a stationary series. Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

8 Tests for Unit Roots Dickey-Fuller Three Different Cases for Dickey-Fuller Tests From McKinnon s Dickey-Fuller Monte Carlo Estimates 1 y t = β 1 y t 1 + u t (no intercept, random walk) Stata: dfuller y,noconstant 2 y t = β 0 + β 1 y t 1 + u t (with intercept, drift term) Stata: dfuller y 3 y t = β 0 + β 1 y t 1 + β 2 t + u t (with drift and a trend on the difference) Stata: dfuller y,trend R, forms #1-3 all at once: adf.test(y) (library: atsa) Testing Dickey-Fuller We can compute a t for β 1 in the normal way, but it is not t-distributed. Its actual distribution is theoretically unknown. But we have Monte Carlo estimates of it for each of the three cases. Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

9 Tests for Unit Roots Dickey-Fuller What Do These Three Cases Look Like? Figure source: CrossValidated user javlacalle ( difference-between-series-with-drift-and-series-with-trend) Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

10 Tests for Unit Roots Dickey-Fuller Augmented Dickey-Fuller Test In addition to a standard Dickey-Fuller test there is also an Augmented Dickey-Fuller (ADF) which additionally tests for autoregressive error of specified lag. R: adf.test in atsa gives the regular version and alternatives at various lags Stata: dfuller varname, lags(#) Software reports an approximate p value for the null hypothesis that the series has a unit root. It is only approximate because the true distribution is unknown. The approximation comes from Monte Carlo estimation. Significance means that the unit root hypothesis is rejected, i.e., that the series is stationary. Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

11 Tests for Unit Roots KPSS Test Kwiatkowski, Perron, Schmidt, and Shinn (KPSS) Test Let us be clear: The null hypothesis is stationary. A significant result means we have a nonstationary series. (Reverse of Dickey-Fuller.) Consider this model: y t = βt + η t + a t. η t = η t 1 + µ t. µ t N (0, σµ). 2 a t N (0, 1). Our null hypothesis is H 0 : σµ 2 = 0. If that is true, then our series is trend stationary (or level stationary if β = 0). This means our alternative hypothesis is H A : σµ 2 > 0. If that is true, then our series has a unit root. Procedure: 1 Regress y t against an intercept and a trend (or just an intercept if you want level-stationarity). 2 Save the residuals e t. 3 Compute the statistic: η j = T 2 T t=1 S2 t st 2 (l) S t is the sum of residuals up to time t. st 2 (l) = T 1 T t=1 e2 t + 2T 1 l s=1 w(s, l) T t=s+1 e te t s is a measure of the error variance of regression. w(s, l) = 1 s l+1 where l is the lag truncation parameter. Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

12 Modeling The Empirical Signature of an Integrated Series We also can turn the the autocorrelation function and partial autocorrelation function to determine if a series has a unit root. I(1) in shock form z t = a t + a t 1 + a t 2 + a t a 0 Note difference from AR(1): There are no decay parameters on the old shocks. Their influence persists forever. We call integrated series permanent memory series. Cookbook Rules for Identification AR(P) Exponential decay in the ACF, P significant spikes in the PACF I(1) Slow decay in the ACF, 1 significant spike in the PACF MA(Q) Q significant spikes in the ACF, exponential decay in the PACF Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

13 Modeling Approaches to Modeling an Integrated Series You Have to Choose the Right Approach for Your Situation Box-Jenkins: If it is integrated, difference it. Once differenced, see if you have any other ARMA processes going on. Prioritize accordingly: y t Integration Filter AR Filter MA Filter a t Observed Stochastic Trend Long Term Short Term White Noise With a final ARIMA model in place, you can model predictors as we did before. Econometric modeling with differenced series. Consider the difference equations and whether you can derive a specification that works. Error Correction Modeling Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14

14 Homework For Next Time Examine the data on short and long term interest rates (RATES.DTA). For both tbonds and prime report the following: Graphs of the ACF and PACF. The results of a Dickey-Fuller test. The results of a KPSS test. Your conclusion on each series: Is it stationary? Estimate an ARIMA model for long-term rates (tbonds). Reading: Time Series Analysis for the Social Sciences, Chapter 6. Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, / 14