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Properties of time series: Lecture 3
How do we identify non-stationary processes?
(A) Informal methods
Plot time series
Correlogram
(B) Formal Methods
Statistical test for stationarity.
Dickey-Fuller tests.
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Thomas 14.2
Informal Procedures to identify non-stationary processes
(1) Eye ball the data (a) Constant mean?
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(b) Constant variance?
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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.
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Informal Procedures to identify non-stationary processes
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Diagnostic test - Correlogram
For a random walk the correlogram does not die out.
High autocorrelation for large values of k
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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
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)
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
ΔXt = α + φ*Xt-1 + ut
ut ~ iid(0,σ2)
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.
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.
Augmented Dickey Fuller (ADF) test for unit root
Dickey Fuller tests assume that the residual ut in eqtn. (2) are nonautocorrelated.
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.
Incorporating time trends in ADF test for unit root
From before some time series clearly display an upward trend (nonstationary 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
It may be the case that Xt will be stationary around a trend.
Although if a trend is not included series is non-stationary.
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Different DF tests – Summary t-type test
ττ
ΔXt = α + βtrend + φ*Xt-1 + ut
(a) Ho: φ* = 0
Ha: φ* < 0
τμ
ΔXt = α + φ*Xt-1 + ut
(b) Ho: φ* = 0
Ha: φ* < 0
τ
ΔXt = φ*Xt-1 + ut
(c) Ho: φ* = 0
Ha: φ* < 0
Critical values from Fuller (1976)
Different DF tests – Summary F-type test
Φ3
ΔXt = α + βtrend + φ*Xt-1 + ut
(a) Ho: φ* = β = 0
Ha: φ* 0 or β 0
Φ1
ΔXt = α + φ*Xt-1 + ut
(b) Ho: φ* = α = 0
Ha: φ* 0 or α 0
Critical values from Dickey and Fuller (1981)
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.