CS代考 Slides-04 Time Series Analysis using ARMA models: Part 2

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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