与Multiplicativ估计回归模型e ARIMA Errors

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This example shows how to fit a regression model with multiplicative ARIMA errors to data usingestimate.

Load the airline and recession data sets. Plot the monthly passenger totals and the log of the totals.

load('Data_Airline.mat') loadData_Recessionsy = Data; logY = log(y); figure subplot(2,1,1) plot(y) title('{\bf Monthly Passenger Totals (Jan1949 - Dec1960)}') datetick subplot(2,1,2) plot(log(y)) title('{\bf Monthly Passenger Log-Totals (Jan1949 - Dec1960)}') datetick

Figure contains 2 axes objects. Axes object 1 with title blank M o n t h l y blank P a s s e n g e r blank T o t a l s blank ( J a n 1 9 4 9 blank - blank D e c 1 9 6 0 ) contains an object of type line. Axes object 2 with title blank M o n t h l y blank P a s s e n g e r blank L o g - T o t a l s blank ( J a n 1 9 4 9 blank - blank D e c 1 9 6 0 ) contains an object of type line.

The log transformation seems to linearize the time series.

Construct the predictor (X), which is whether the country was in a recession during the sampled period. A 0 in rowtmeans the country was not in a recession in montht, and a 1 in rowtmeans that it was in a recession in montht.

X = zeros(numel(dates),1);% Preallocationforj = 1:size(Recessions,1) X(dates >= Recessions(j,1) & dates <= Recessions(j,2)) = 1;end

Fit the simple linear regression model

$y_{t} = c + X_{t} β + u_{t}$

to the data.

Fit = fitlm(X,logY);

Fitis aLinearModelthat contains the least squares estimates.

Check for standard linear model assumption departures by plotting the residuals several ways.

figure subplot(2,2,1) plotResiduals(Fit,'caseorder','ResidualType','Standardized',...'LineStyle',“- - -”,'MarkerSize',0.5) h = gca; h.FontSize = 8; subplot(2,2,2) plotResiduals(Fit,'lagged','ResidualType','Standardized') h = gca; h.FontSize = 8; subplot(2,2,3) plotResiduals(Fit,'probability','ResidualType','Standardized') h = gca; h.YTick = h.YTick(1:2:end); h.YTickLabel = h.YTickLabel(1:2:end,:); h.FontSize = 8; subplot(2,2,4) plotResiduals(Fit,'histogram','ResidualType','Standardized') h = gca; h.FontSize = 8;

Figure contains 4 axes objects. Axes object 1 with title Case order plot of residuals contains 2 objects of type line. Axes object 2 with title Plot of residuals vs. lagged residuals contains 3 objects of type line. Axes object 3 with title Normal probability plot of residuals contains 2 objects of type line. Axes object 4 with title Histogram of residuals contains an object of type patch.

r = Fit.Residuals.Standardized; figure subplot(2,1,1) autocorr(r) h = gca; h.FontSize = 9; subplot(2,1,2) parcorr(r) h = gca; h.FontSize = 9;

Figure contains 2 axes objects. Axes object 1 with title Sample Autocorrelation Function contains 4 objects of type stem, line. Axes object 2 with title Sample Partial Autocorrelation Function contains 4 objects of type stem, line.

The residual plots indicate that the unconditional disturbances are autocorrelated. The probability plot and histogram seem to indicate that the unconditional disturbances are Gaussian.

The ACF of the residuals confirms that the unconditional disturbances are autocorrelated.

Take the 1st difference of the residuals and plot the ACF and PACF of the differenced residuals.

博士= diff (r);图次要情节(2,1,1)autocorr (dR,'NumLags',50) h = gca; h.FontSize = 9; subplot(2,1,2) parcorr(dR,'NumLAgs',50) h = gca; h.FontSize = 9;

The ACF shows that there are significantly large autocorrelations, particularly at every 12th lag. This indicates that the unconditional disturbances have 12th degree seasonal integration.

Take the first and 12th differences of the residuals. Plot the differenced residuals, and their ACF and PACF.

DiffPoly = LagOp([1 -1]); SDiffPoly = LagOp([1 -1],'Lags',[0, 12]); diffR = filter(DiffPoly*SDiffPoly,r); figure subplot(2,1,1) plot(diffR) axistightsubplot(2,2,3) autocorr(diffR) h = gca; h.FontSize = 7; axistightsubplot(2,2,4) parcorr(diffR) h = gca; h.FontSize = 7; axistight

Figure contains 3 axes objects. Axes object 1 contains an object of type line. Axes object 2 with title Sample Autocorrelation Function contains 4 objects of type stem, line. Axes object 3 with title Sample Partial Autocorrelation Function contains 4 objects of type stem, line.

The residuals resemble white noise (with possible heteroscedasticity). According to Box and Jenkins (1994), Chapter 9, the ACF and PACF indicate that the unconditional disturbances are an $I M A (0, 1, 1) \times (0, 1, 1)_{12}$ model.

Specify the regression model with $I M A (0, 1, 1) \times (0, 1, 1)_{12}$ errors:

$\begin{array}{c} y_{t} = X_{t} β + u_{t} \\ (1 - L) (1 - L^{12}) u_{t} = (1 + b_{1} L) (1 + B_{12} L^{12}) ε_{t} . \end{array}$

Mdl = regARIMA('MALags',1,'D',1,'Seasonality',12,'SMALags',12)

Mdl = regARIMA with properties: Description: "ARIMA(0,1,1) Error Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution)" Distribution: Name = "Gaussian" Intercept: NaN Beta: [1×0] P: 13 D: 1 Q: 13 AR: {} SAR: {} MA: {NaN} at lag [1] SMA: {NaN} at lag [12] Seasonality: 12 Variance: NaN

Partition the data set into the presample and estimation sample so that you can initialize the series.P=Q= 13, so the presample should be at least 13 periods long.

preLogY = logY(1:13);% Presample responsesestLogY = logY(14:end);% Estimation sample responsespreX = X(1:13);% Presample predictorsestX = X(14:end);% Estimation sample predictors

Obtain presample unconditional disturbances from a linear regression of the presample data.

PreFit = fitlm(preX,preLogY);...% Presample fit for presample residualsEstFit = fitlm(estX,estLogY);...% Estimation sample fit for the interceptU0 = PreFit.Residuals.Raw;

If the error model is integrated, then the regression model intercept is not identifiable. SetInterceptto the estimated intercept from a linear regression of the estimation sample data. Estimate the regression model with IMA errors.

Mdl.Intercept = EstFit.Coefficients{1,1}; EstMdl = estimate(Mdl,estLogY,'X',estX,'U0',U0);

Regression with ARIMA(0,1,1) Error Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Intercept 5.5722 0 Inf 0 MA{1} -0.025366 0.22197 -0.11427 0.90902 SMA{12} -0.80255 0.052705 -15.227 2.3349e-52 Beta(1) 0.0027588 0.10139 0.02721 0.97829 Variance 0.0072463 0.00015974 45.365 0

MA{1}andBeta1are not significantly different from 0. You can remove these parameters from the model, possibly add other parameters (e.g., AR parameters), and compare multiple model fits usingaicbic. Note that the estimation and presample should be the same over competing models.

References:

Box, G. E. P., G. M. Jenkins, and G. C. Reinsel.Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.

与Multiplicativ估计回归模型e ARIMA Errors

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