1. How do we identify non-stationary processes? (A) Informal methods Thomas 14.1 Plot time series Correlogram (B) Formal Methods Statistical test for stationarity. Thomas 14.2 Dickey-Fuller tests. Properties of time series: Lecture 3

2. Informal Procedures to identify non-stationary processes (1) Eye ball the data (a) Constant mean? (b) Constant variance?

3. Informal Procedures to identify non-stationary processes (2)Diagnostic test - Correlogram Correlation between 1980 and 1980 + k. For stationary process correlogram dies out rapidly. Series has no memory. 1980 is not related to 1985.

4. Informal Procedures to identify non-stationary processes (2)Diagnostic test - Correlogram For a random walk the correlogram does not die out. High autocorrelation for large values of k

5. Statistical Tests for stationarity: Simple t-test Set up AR(1) processwith drift (α) X t = α + φX t-1 + u t u t ~ iid(0,σ 2 ) (1) Simple approach is to estimate eqtn (1) using OLS and examine estimated φ {phi} Use a t-test with null Ho: φ = 1 (non-stationary) against alternative Ha: φ < 1 (stationary). Test Statistic: TS = (φ – 1) / (Std. Err.(φ)) reject null hypothesis when test statistic is large negative - 5% critical value is -1.65

6. Statistical Tests for stationarity: Simple t-test Simple t-test based on AR(1) processwith drift (α) X t = α + φX t-1 + u t u t ~ iid(0,σ 2 ) (1) Problem with simple t-test approach (1)lagged dependent variables => φ biased downwards in small samples (i.e. dynamic bias) (2)When φ =1, we have non-stationary process and standard regression analysis is invalid (i.e. non-standard distribution)

7. Dickey Fuller (DF) approach to non-stationarity testing Dickey and Fuller (1979) suggest we subtract X t-1 from both sides of eqtn. (1) X t - X t-1 = α + φX t-1 - X t-1 + u t u t ~ iid(0,σ 2 ) ΔX t = α + φ*X t-1 + u t φ* = φ –1 (2) Use a t-test with: null Ho: φ* = 0 (non-stationary or Unit Root) against alternative Ha: φ* < 0 (stationary). - Large negative test statistics reject non-stationarity - This is know as unit root test since in eqtn. (1) Ho: φ =1.

8. Dickey Fuller (DF) tests for unit root Use t-test with a non-standard distribution because of (1) dynamic bias in eqtn (1) (2) non-stationary variables under null - distribution of Dickey-Fuller test statistics – created by simulation critical value for τ μ -test are larger than normal t-test. τ {tau} Example Sample size of n = 25 at 5% level of significance for eqtn. (2) τ μ -critical value = -3.00t-test critical value = -1.65 Δp t-1 = -0.007 - 0.190p t-1 (-1.05) (-1.49) φ* = -0.190τ μ = -1.49 > -3.00hence unit root.

9. Augmented Dickey Fuller (ADF) test for unit root Dickey Fuller tests assume that the residual u t in eqtn. (2) are non- autocorrelated. Solution: incorporate lagged dependent variables. For quarterly data add up to four lags. ΔX t = α + φ*X t-1 + θ 1 ΔX t-1 + θ 2 ΔX t-2 + θ 3 ΔX t-3 + θ 4 ΔX t-4 + u t (3) Problem arises of differentiating between models. Use a general to specific approach to eliminate insignificant variables Check final parsimonious model for autocorrelation. Check F-test for significant variables Use Information Criteria. Trade-off parsimony vs residual variance.

10. Incorporating time trends in ADF test for unit root From before some time series clearly display an upward trend (non- stationary mean). Should therefore incorporate trend in ADF test (i.e. equation 3). ΔX t = α + βtrend + φ*X t-1 + θ 1 ΔX t-1 + θ 2 ΔX t-2 + θ 3 ΔX t-3 + θ 4 ΔX t-4 + u t (4) It may be the case that X t will be stationary around a trend. Although if a trend is not included series is non-stationary.

11. Different DF tests – Summary t-type test τ τ ΔX t = α + βtrend + φ*X t-1 + u t (a)Ho: φ* = 0Ha: φ* < 0 τ μ ΔX t = α + φ*X t-1 + u t (b)Ho: φ* = 0Ha: φ* < 0 τ ΔX t = φ*X t-1 + u t (c) Ho: φ* = 0Ha: φ* < 0 Critical values from Fuller (1976)

12. Different DF tests – Summary F-type test Φ 3 ΔX t = α + βtrend + φ*X t-1 + u t (a)Ho: φ* = β = 0Ha: φ*  0 or β  0 Φ 1 ΔX t = α + φ*X t-1 + u t (b)Ho: φ* = α = 0Ha: φ*  0 or α  0 Critical values from Dickey and Fuller (1981)

13. Alternative statistical test for stationarity One further approach is the Sargan and Bhargava (1983) test which uses the Durbin-Watson statistic. If X t is regressed on a constant alone, we then examine the residuals for serial correlation. Serial correlation in the residuals (long memory) will fail the DW test and result in a low value for this test. This test has not proven so popular.