Slides-04 Time Series Analysis using ARMA models: Part 2
Univariate Time Series Analysis: ARIMA models
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Fitting ARMA models to the data
Estimating ARMA models
Example: ∆ ln(GDP) for Belgium
→ estimate tentative models
Figure 50 : Estimated AR(1) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Estimating ARMA models
Figure 51 : Estimated AR(1) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Estimating ARMA models
Figure 52 : Estimated AR(2) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Estimating ARMA models
Figure 53 : Estimated AR(3) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Estimating ARMA models
Figure 54 : Estimated MA(1) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Estimating ARMA models
Figure 55 : Estimated MA(2) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Estimating ARMA models
Figure 56 : Estimated ARMA(1,1) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Estimating ARMA models
Figure 57 : Estimated ARMA(1,2) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Estimating ARMA models
Figure 58 : Estimated ARMA(2,1) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Estimating ARMA models
Figure 59 : Estimated ARMA(2,2) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Estimating ARMA models
Figure 60 : Estimated ARMA(3,2) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
Example: ∆ ln(GDP) for Belgium
I Overfitting: test e.g. the joint significance of the
MA-coefficients when going from the AR(3) to the
ARMA(3,2) model
(0.002077− 0.001962) /2
0.001962 /(147− 6)
where the 5% critical values ≈ 3.07.
Or test the joint significance of the coefficients needed for
going from the ARMA(3,2) to the ARMA(4,4) model
(0.001962− 0.001630) /3
0.001630 /(147− 9)
where the 5% critical values ≈ 2.68.
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
Figure 62 : Estimated ARMA(4,4) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
I Residual diagnostics: note that for both the AR(3) and the
ARMA(3,2) there is autocorrelation left in the residuals.
This indicates/implies that:
I The fitted ARMA models are not rich enough to capture all of
the dynamics in ∆ ln(GDP) for Belgium
I The least squares estimator is biased and inconsistent!
Note that especially the correlation at lag 4 looks significant.
As we have quarterly data, this might be a seasonal effect. In
order to account for seasonality, an additional MA coefficient
at lag 4 is added. For truly seasonal patterns, such an
MA-component best captures spikes (and not decay) at the
quarterly lags. Also note that the MA(4) is highly significant
in the ARMA(4,4) model.
I An ARMA(1,(2,4)) model has the smallest AIC and SBC with
the residuals being ≈ white noise.
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
Figure 63 : Correlogram estimated residuals from AR(3) model for
∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
Figure 64 : Correlogram estimated residuals from ARMA(3,2) model for
∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
Figure 65 : Estimated ARMA(1,(2,4)) model for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
Figure 66 : Correlogram estimated residuals from ARMA(1,(2,4)) model
for ∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
Figure 67 : Impulse response function for estimated AR(3), ARMA(3,2)
and ARMA(1,(2,4)) model for ∆ ln(GDP) Belgium
t-5 t t+5 t+10 t+15 t+20 t+25
ARMA(1,(2,4))
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
I Parameter stability test: note that the DGP appears to
change around 1995.
Split the sample in two sub-samples, e.g. 1970:1-1994:4 and
1995:1-2007:4, and perform Chow test.
(0.001650− (0.000539 + 0.000809)) /4
(0.000539 + 0.000809) /(147− 8)
where the 5% critical values ≈ 2.45.
ARMA process is not stable over the sample period!
Especially if you want to predict future output growth, you
better estimate the ARMA process over a smaller sample size
in order to avoid parameter instability. The model estimated
over the period 1995:1-2007:4 passes the diagnostic checks,
i.e. no autocorrelation in the residuals and no parameter
instability (check!).
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
Figure 68 : Estimated residuals from ARMA(1,(2,4)) model for
∆ ln(GDP) Belgium
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
Figure 69 : Estimated ARMA(1,(2,4)) model for ∆ ln(GDP) Belgium
(1971:2-1994:4)
Univariate Time Series Analysis: ARIMA models
Fitting ARMA models to the data
Diagnostic Checking
Figure 70 : Estimated ARMA(1,(2,4)) model for ∆ ln(GDP) Belgium
(1995:1-2007:4)
Univariate Time Series Analysis: ARIMA models
Forecasting using ARMA models
Building Forecasts
Figure 72 : Using an MA(2) model to forecast ∆ ln(GDP) from 2005:1
onward (estimation period: 1995:1-2004:4)
2000/Q1 2001/Q1 2002/Q1 2003/Q1 2004/Q1 2005/Q1 2006/Q1 2007/Q1
Dynamic forecast
Static forecast
Univariate Time Series Analysis: ARIMA models
Forecasting using ARMA models
Building Forecasts
Figure 73 : Using an AR(1) model to forecast ∆ ln(GDP) from 2005:1
onward (estimation period: 1995:1-2004:4)
2000/Q1 2001/Q1 2002/Q1 2003/Q1 2004/Q1 2005/Q1 2006/Q1 2007/Q1
Dynamic forecast
Static forecast
Univariate Time Series Analysis: ARIMA models
Forecasting using ARMA models
Building Forecasts
Figure 74 : Using an ARMA(1,(2,4)) model to forecast ∆ ln(GDP) from
2005:1 onward (estimation period: 1995:1-2004:4)
2000/Q1 2001/Q1 2002/Q1 2003/Q1 2004/Q1 2005/Q1 2006/Q1 2007/Q1
Dynamic forecast
Static forecast
Univariate Time Series Analysis: ARIMA models
Forecasting using ARMA models
Forecasting Accuracy
Figure 75 : Using an MA(2) model to forecast ∆ ln(GDP) from 2005:1
onward (estimation period: 1995:1-2004:4)
2000/Q1 2001/Q1 2002/Q1 2003/Q1 2004/Q1 2005/Q1 2006/Q1 2007/Q1
Dynamic forecast
+/- 2 s.e.
Univariate Time Series Analysis: ARIMA models
Forecasting using ARMA models
Forecasting Accuracy
Figure 76 : Using an AR(1) model to forecast ∆ ln(GDP) from 2005:1
onward (estimation period: 1995:1-2004:4)
2000/Q1 2001/Q1 2002/Q1 2003/Q1 2004/Q1 2005/Q1 2006/Q1 2007/Q1
Dynamic forecast
+/- 2 s.e.
Univariate Time Series Analysis: ARIMA models
Forecasting using ARMA models
Forecasting Accuracy
Figure 77 : Using an ARMA(1,(2,4)) model to forecast ∆ ln(GDP) from
2005:1 onward (estimation period: 1995:1-2004:4)
2000/Q1 2001/Q1 2002/Q1 2003/Q1 2004/Q1 2005/Q1 2006/Q1 2007/Q1
Dynamic forecast
+/- 2 s.e.
Univariate Time Series Analysis: ARIMA models
Forecasting using ARMA models
Forecasting Accuracy
Figure 78 : Forecast accuracy of an MA(2) model in forecasting
∆ ln(GDP) from 2005:1 onward (estimation period: 1995:1-2004:4)
Univariate Time Series Analysis: ARIMA models
Forecasting using ARMA models
Forecasting Accuracy
Figure 79 : Forecast accuracy of an AR(1) model in forecasting
∆ ln(GDP) from 2005:1 onward (estimation period: 1995:1-2004:4)
Univariate Time Series Analysis: ARIMA models
Forecasting using ARMA models
Forecasting Accuracy
Figure 80 : Forecast accuracy of an ARMA(1,(2,4)) model in forecasting
∆ ln(GDP) from 2005:1 onward (estimation period: 1995:1-2004:4)
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