Unit Roots & Forecasting

1. Unit Roots & Forecasting Methods of Economic Investigation Lecture 20

2. Last Time • Descriptive Time Series • Processes • Estimating with exogenous serial correlation • Estimating with endogenous processes

3. Today’s Class • Non-stationaryTime Series • Unit Roots and Spurious Regressions • Orders of Integration • Returning to Causal Effects • Impulse Response Functions • Forecasting

4. Random Walk Processes • Definition: • Et[xt+1] = xt that is today’s value of X is the best predictor of tomorrow’s value. • This looks very similar to our AR(1) processes, where φ = 1. • Autocovariances of a random walk are not well defined in a technical sense, but imagine AR(1) process with φ1: we have nearly perfect autocorrelation for any two time periods. • persistence dies out too slowly so most of variance will largely be due to very low-frequency “shocks.”

5. Permanence of Shocks in Unit Root • An innovation (a shock at t ) to a stationary AR process dies out eventually (the autocorrelation function declines eventually to zero). • A shock to a random walk is permanent • Variance is increasing over time • Var(xt) = Var(x0) + tσ2

7. Drifts and Trends • Deterministic trend • yt = δt + xt + εt • xt is some stationary process • yt is “trend” stationary • It’s easy to add a deterministic trend to a random walk

9. Orders of Integration • A series is integrated of order p if a p differences render it stationary. • If a time series is integrated and differencing once renders the time series stationary, then it is integrated of order 1 or I(1). • If it is necessary to difference twice before a time series is stationary, then it is I(2), and so forth.

10. Integrated Series • If a time series has a unit root, it is said to be integrated. • First differencing the time series removes the unit root. E.g. in the case of a random walk yt = yt-1 + ut, ut ~ N(0, σ2) Δyt = ut • the first difference is white noise, which is stationary. • For an AR(p) a unit root implies 1 – β1L – β2L2 – ... – βpLp = (1 – L) (1 – λ1L – λ2L2 ... – λpLp-1)= 0 • and as a result first differencing also removes the unit root.

11. Non-stationarity • Nonstationarity can have important consequences for regression modelsand inference. • Autoregressive coefficients are biased • t-statistics have non-normal distributions even in large samples • Spurious regression

12. Problem: Spurious Regression imagine we now have two series are generated by independent random walks, Suppose we run yt on xt using OLS, that is we estimate yt = α + βxt + νt. In this case, you tend to see ”significant” β because the low-frequency changes make it seem as if the two series are in some way associated.

14. Unit Root Tests • Standard Dickey-Fuller test appropriate for AR(1) processes • Many economic and financial time series have a more complicated dynamic structure than is captured by a simple AR(1) model. • Said and Dickey (1984) augment the basic autoregressive unit root test to accommodate general ARMA(p, q) models with unknown orders and • Called the augmented Dickey-Fuller (ADF) test

15. ADF Test – 1 • The ADF test tests the null hypothesis that a time series yt is I(1) against the alternative that it is I(0), assuming that the dynamics in the data have an ARMA structure. • The ADF test is based on estimating the test regression Other serial correlation Deterministic variables Potential unit root

16. ADF Test - 2 • To see why: • Subtract yt-1 from both sides and define Φ = (α1+ α2+…+ αp – 1)and we get • Test Φ= 0 against alternative Φ<0 • Use special DF upperbound and lowerbound • Under alternative, test statistic is not normally distributed

17. Estimating in Time Series • Non-stationary time series can lead to a lot of problems in econometric analysis. • In order to work with time series, particular in regression models, we should therefore transform our variables to stationary time series first. • First differencing removes unit roots or trends. Hence, difference a time series until it is I(0). • Differencing too often is less of a problem since a differenced stationary series is still stationary. • Regressions of one stationary variable on another is less problematic. • Although observations may not be independent, we can expect regression to have similar properties as with cross sectional data.

18. Impulse Response Function • One of the most interesting things to do with an ARMA model is form predictions of the variable given its past. • we want to know what is Et(xt+j ) • Can do inference with Vart(xt+j) • The impulse response function is a simpel way to do that • Follow te path that x follows if it is kicked by unit shock • characterization of the behavior of our models. • allows us to start thinking about “causes” and “effects”.

19. Impulse Response and MA(∞) • 1. The MA(∞) representation is the same thing as the impulse response function. i.e. • The easiest way to calculate an MA(∞) representation is to simulate the impulse-response function. • The impulse response function is the same as Et(xt+j) − Et−1(xt+j).

20. Causality and Impulse Response • Can either forecast or simulate the effect of a given shock • Try to pick a shock time/level to simulate and try to replicate observed data • Issue of whether that shock is what really happened • Know a shock happened in time time t • See if observed change (more on this next time) • Granger causality implies a correlation between the current value of one variable and the past values of others • it does not necessarily imply that changes in one variable “causes” changes in another. • Use a F-test to jointly test for the significance of the lags on the explanatory variables, this in effect tests for ‘Granger causality’ between these variables. • Can visually see correlation in impulse response functions

21. Source: Cochrane, QJE (1994)

22. Next Time • Estimating Causality in Time Series • Some additional forecasting stuff • Testing for breaks • Regression discontinuity/Event study