Economics 310 Lecture 25 Univariate Time-Series

Methods of Economic Forecasting Single-equation regression models Simultaneous-equation regression models. Autoregressive integrated moving average (ARIMA) models. Vector autoregressive (VAR) models.

Justification for ARIMA models Lucas critique Parameters estimated from an econometric model are dependent on the policy prevailing at the time the model was estimated. They will change if the policy changes. Let the data speak for themselves.

An Autoregressive (AR) Process

Moving Average (MA) process

ARMA

Autoregressive Integrated Moving Average Model To use an ARMA model, the time-series must be stationary. Many series must be integrated (differenced) to make them stationary. We write these series as I(d), where d=number of differences needed to get stationarity. If we model the I(d) series as an ARMA(p,q) model, we get an ARIMA(p,d,q) model. Where p=degree of autoregressive model, d=degree of integration and q=degree of moving average term.

Box-Jenkins Methodology Identification-find the values of p,d,q for series. Estimation-how we estimate parameters of the ARIMA model. Diagnostic checking-How well does the model fit the series. Forecasting-Good for short-term forecasting

Identification We will use the autocorrelation function (ACF) and the partial autocorrelation function (PACF). Partial autocorrelation function measures correlation between (time-series) observations that are k time periods apart after controlling for correlations at intermediate lags (lags less than k).

Theoretical patterns of ACF & PACF Type of ModelTypical Pattern of ACF Typical Pattern of PACF AR(p) Decays exponentially or with damped sine wave pattern or both. Significant spikes through lags p. MA(q) Significant spikes through lags q. Declines exponentially. ARMA(p,q) Exponential Decay.

Correlogram for Exchange Rate AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) TWEX RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRR.

Partial Correlogram for Exchange Rate PARTIAL AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) TWEX RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRR RR RRRR R R R RRR RRRR RR RRRR RRR R RR RRR RR R RRRRR.

Dickey-Fuller Test on Exchange Rate VARIABLE : TWEX DICKEY-FULLER TESTS - NO.LAGS = 17 NO.OBS = 273 NULL TEST ASY. CRITICAL HYPOTHESIS STATISTIC VALUE 10% CONSTANT, NO TREND A(1)=0 T-TEST A(0)=A(1)= AIC = SC = CONSTANT, TREND A(1)=0 T-TEST A(0)=A(1)=A(2)= A(1)=A(2)= AIC = SC =

Correlogram for 1 st Difference of Exchange Rate AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) TWEXDIFF RRRRRRRRRRR RR RRRR RR R R RR RRRRR RRR RRRR RRRR RRR R RR RR R RRRRR R +.

Partial Correlogram for 1 st Difference of Exchange Rate PARTIAL AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) TWEXDIFF RRRRRRRRRRR RRR RRRR RR R R RR RRRR R RRR RR R R RR R RR RRRRRR RRR +.

Estimation of ARIMA(1,1,0) NET NUMBER OF OBS IS 290 DIFFERENCING: 0 CONSECUTIVE, 0 SEASONAL WITH SPAN 0 CONVERGENCE AFTER 11 ITERATIONS INITIAL SUM OF SQS= FINAL SUM OF SQS= R-SQUARE = R-SQUARE ADJUSTED = VARIANCE OF THE ESTIMATE-SIGMA**2 = STANDARD ERROR OF THE ESTIMATE-SIGMA = AKAIKE INFORMATION CRITERIA -AIC(K) = SCHWARZ CRITERIA- SC(K) = PARAMETER ESTIMATES STD ERROR T-STAT AR( 1) E CONSTANT E E

Estimation of ARIMA(0,1,1) NET NUMBER OF OBS IS 290 DIFFERENCING: 0 CONSECUTIVE, 0 SEASONAL WITH SPAN 0 CONVERGENCE AFTER 13 ITERATIONS INITIAL SUM OF SQS= FINAL SUM OF SQS= R-SQUARE = R-SQUARE ADJUSTED = VARIANCE OF THE ESTIMATE-SIGMA**2 = STANDARD ERROR OF THE ESTIMATE-SIGMA = AKAIKE INFORMATION CRITERIA -AIC(K) = SCHWARZ CRITERIA- SC(K) = PARAMETER ESTIMATES STD ERROR T-STAT MA( 1) E CONSTANT E

Estimation of ARIMA(1,1,1) NET NUMBER OF OBS IS 290 DIFFERENCING: 0 CONSECUTIVE, 0 SEASONAL WITH SPAN 0 CONVERGENCE AFTER 14 ITERATIONS INITIAL SUM OF SQS= FINAL SUM OF SQS= R-SQUARE = R-SQUARE ADJUSTED = VARIANCE OF THE ESTIMATE-SIGMA**2 = STANDARD ERROR OF THE ESTIMATE-SIGMA = AKAIKE INFORMATION CRITERIA -AIC(K) = SCHWARZ CRITERIA- SC(K) = PARAMETER ESTIMATES STD ERROR T-STAT AR( 1) E MA( 1) CONSTANT E

Estimation of ARIMA(1,1,0) with seasonal of 17 NET NUMBER OF OBS IS 290 DIFFERENCING: 0 CONSECUTIVE, 0 SEASONAL WITH SPAN 17 CONVERGENCE AFTER 12 ITERATIONS INITIAL SUM OF SQS= FINAL SUM OF SQS= R-SQUARE = R-SQUARE ADJUSTED = VARIANCE OF THE ESTIMATE-SIGMA**2 = STANDARD ERROR OF THE ESTIMATE-SIGMA = AKAIKE INFORMATION CRITERIA -AIC(K) = SCHWARZ CRITERIA- SC(K) = PARAMETER ESTIMATES STD ERROR T-STAT AR( 1) E SAR( 1) E CONSTANT E E

Estimation of ARIMA(0,1,1) with seasonal of 17 NET NUMBER OF OBS IS 290 DIFFERENCING: 0 CONSECUTIVE, 0 SEASONAL WITH SPAN 17 CONVERGENCE AFTER 14 ITERATIONS INITIAL SUM OF SQS= FINAL SUM OF SQS= R-SQUARE = R-SQUARE ADJUSTED = VARIANCE OF THE ESTIMATE-SIGMA**2 = STANDARD ERROR OF THE ESTIMATE-SIGMA = AKAIKE INFORMATION CRITERIA -AIC(K) = SCHWARZ CRITERIA- SC(K) = PARAMETER ESTIMATES STD ERROR T-STAT MA( 1) E SMA( 1) E CONSTANT E

Selecting a Model ModelAICSCHWARZ ARIMA(1,1,0) ARIMA(0,1,1) ARIMA(1,1,1) ARIMA(1,1,0) Seasonal ARIMA(0,1,1) Seasonal

Diagnostic We want to check the adequacy of our model. For an ARIMA(p,d,q), check that the added coefficients for ARIMA(p+1,d,q) and ARIMA(p,d,q+1) are zero. Do a plot of the autocorrelation of residuals from the model to see that they are white noise. Run a Ljung-Box test on the residuals to see that they are White noise.

Diagnostics-ARIMA(0,1,1) 17 month seasonal effect MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE LAG Q DF P-VALUE

Diagnostics-ARIMA(0,1,1) 17 month seasonal effect RESIDUALS LAGS AUTOCORRELATIONS STD ERR

Diagnostics-ARIMA(0,1,1) 17 month seasonal effect AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) LACKFIT R RR RRR RR R R R RRRRR RR RRR RRR RR R RRR RR RR R RRR +.

Diagnostics-ARIMA(0,1,1) 17 month seasonal effect PARTIAL AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) LACKFIT R RR RRR RR R R R RRRRR RR RRR RR RR +.

Diagnostics-ARIMA(0,1,1) 17 month seasonal effect PARAMETER ESTIMATES STD ERROR T-STAT MA( 1) E MA( 2) E E E-01 SMA( 1) E CONSTANT E RESIDUALS LAGS AUTOCORRELATIONS STD ERR