1 / 13

How do we identify non-stationary processes? (A) Informal methods Thomas 14.1

Properties of time series: Lecture 3. 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.

ivria
Télécharger la présentation

How do we identify non-stationary processes? (A) Informal methods Thomas 14.1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Properties of time series: Lecture 3 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.

  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) process with drift (α) Xt = α + φXt-1 + ut ut ~ iid(0,σ2) (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) process with drift (α) • Xt = α + φXt-1 + ut ut ~ iid(0,σ2) (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 Xt-1 from both sides of eqtn. (1) • Xt - Xt-1 = α + φXt-1 - Xt-1 + ut ut ~ iid(0,σ2) • ΔXt = α + φ*Xt-1 + ut 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 τμ-test are larger than normal t-test. τ {tau} • Example • Sample size of n = 25 at 5% level of significance for eqtn. (2) • τμ-critical value = -3.00 t-test critical value = -1.65 • Δpt-1 = -0.007 - 0.190pt-1 • (-1.05) (-1.49) • φ* = -0.190 τμ = -1.49 > -3.00 hence unit root.

  9. Augmented Dickey Fuller (ADF) test for unit root Dickey Fuller tests assume that the residual ut in eqtn. (2) are non-autocorrelated. Solution: incorporate lagged dependent variables. For quarterly data add up to four lags. ΔXt = α + φ*Xt-1 + θ1ΔXt-1 + θ2ΔXt-2 + θ3ΔXt-3 + θ4ΔXt-4 + ut (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). • ΔXt = α + βtrend + φ*Xt-1 • + θ1ΔXt-1 + θ2ΔXt-2 + θ3ΔXt-3 + θ4ΔXt-4 + ut (4) • It may be the case that Xt will be stationary around a trend. Although if a trend is not included series is non-stationary.

  11. Different DF tests – Summaryt-type test ττΔXt = α + βtrend + φ*Xt-1 + ut (a) Ho:φ*= 0Ha:φ*< 0 τμΔXt = α + φ*Xt-1+ ut (b) Ho: φ* = 0 Ha: φ* < 0 τΔXt = φ*Xt-1 + ut (c)Ho: φ* = 0 Ha: φ* < 0 Critical values from Fuller (1976)

  12. Different DF tests – Summary F-type test Φ3ΔXt = α + βtrend + φ*Xt-1 + ut (a) Ho:φ*= β =0Ha:φ* 0 or β0 Φ1 ΔXt = α + φ*Xt-1+ ut (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 Xt 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.

More Related