lab five postscript

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Lab Five Postscript

Econ 240 CEcon 240 C

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Airline Passengers

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Monthly Airline Passengers

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Pathologies of Non-Stationarity

Trend in varianceTrend in variance Trend in meanTrend in mean seasonalseasonal

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Fix-Up: Transformations

Natural logarithmNatural logarithm First difference: (1-Z)First difference: (1-Z) Seasonal difference: (1-ZSeasonal difference: (1-Z1212))

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LNBJPASS

Natural Logarithm of Airline Passengers

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DLNBJPASS

First Difference of Natural Logarithm:Fractional Change in Airline Passengers

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Series: DLNBJPASSSample 1949:02 1960:12Observations 143

Mean 0.009440Median 0.014815Maximum 0.223144Minimum -0.223144Std. Dev. 0.106556Skewness -0.086724Kurtosis 1.947744

Jarque-Bera 6.776578Probability 0.033766

Fractional Change in Airline Passengers

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SDDLNBJP

Seasonal Difference in Fractioanal ChangeIn Airline Passengers

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Series: SDDLNBJPSample 1950:02 1960:12Observations 131

Mean 0.000291Median 0.000000Maximum 0.140720Minimum -0.141343Std. Dev. 0.045848Skewness 0.038191Kurtosis 4.147777


Seasonal Difference in Fractional Changein Airline Passengers

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Proposed Model

Autocorrelation FunctionAutocorrelation Function Negative ACF(1) @ lag oneNegative ACF(1) @ lag one Negative ACF(12) @ lad twelveNegative ACF(12) @ lad twelve

Trial model SDDLNBJP c ma(1) ma(12)Trial model SDDLNBJP c ma(1) ma(12) Sddlnbjp(t) = c + resid(t)Sddlnbjp(t) = c + resid(t) Resid(t) = wn(t)– aResid(t) = wn(t)– a11wn(t-1) – awn(t-1) – a12 12 wn(t-12)wn(t-12)

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Is this model satisfactory? DiagnosticsDiagnostics

Goodness of fit: how well does the model Goodness of fit: how well does the model (fitted value) track the data (observed (fitted value) track the data (observed value)? value)? Plot of actual Vs. fittedPlot of actual Vs. fitted

Is there any structure left in the residual? Is there any structure left in the residual? Correlogram of the residual from the Correlogram of the residual from the model.model.

Is the residual normal? Is the residual normal? Histogram of the Histogram of the residual.residual.

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Residual Actual Fitted

Diagnostics

Plot of actual, fitted, and residual

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Correlogram of the residual from the model

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Series: ResidualsSample 1950:02 1960:12Observations 131

Mean 0.001816Median 0.000352Maximum 0.106210Minimum -0.109547Std. Dev. 0.037828Skewness 0.028306Kurtosis 3.441251


Is the Orthogonal Residual Normally Distributed?

Histogram of the residual

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Forecasting Seasonal Difference in the Fractional Change Estimation period: 1949.01 – 1960.12Estimation period: 1949.01 – 1960.12 Forecast period: 1961.01 – 1961.12Forecast period: 1961.01 – 1961.12

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Eviews forecast command window

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Eviews plot of forecast plus or minus two standard errors Of the forecast

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Eviews spreadsheet view of the forecast and the standard Error of the forecast

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Using the Quick Menu and the show command to create Your own plot or display of the forecast

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SDDLNBJPSDDLNBJPF

SDDLNBJPF+2*SEFSDDLNBJPF-2*SEF

!2 Month Forecadt of Fractional Change

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Note: EViews sets the forecast variable equal to the observedValue for 1949.01-1960.12.

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To Differentiate the Forecast from the observed variable …. In the spread sheet window, click on edit, In the spread sheet window, click on edit,

and copy the forecast values for 1961.01-and copy the forecast values for 1961.01-1961.12 to a new column and paste. Label 1961.12 to a new column and paste. Label this column forecast.this column forecast.

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Note: EViews sets the forecast variable equal to the observedValue for 1949.01-1960.12.

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Displaying the Forecast

Now you are ready to use the Quick menu Now you are ready to use the Quick menu and the show command to make a more and the show command to make a more pleasing display of the data, the forecast, pleasing display of the data, the forecast, and its approximate 95% confidence and its approximate 95% confidence interval.interval.

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Qick menu, show command window

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SDDLNBJPFORECAST

FORECAST+2*SEFFORECAST-2*SEF

Twelve Month Forecast ofSeasonal Difference in Fractional Change

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Recoloring The seasonal difference of the fractional change in The seasonal difference of the fractional change in

airline passengers may be appropriately pre-airline passengers may be appropriately pre-whitened for Box-Jenkins modeling, but it is whitened for Box-Jenkins modeling, but it is hardly a cognitive or intuitive mode for hardly a cognitive or intuitive mode for understanding the data.Fortunately, the understanding the data.Fortunately, the transformation process is reversible and we transformation process is reversible and we recolor, I.e put back the structure we removed recolor, I.e put back the structure we removed with the transformations by using the definitions with the transformations by using the definitions of the transformations themselvesof the transformations themselves

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Recoloring Summation or integration is the opposite of Summation or integration is the opposite of

differencing.differencing. The definition of the first difference is: (1-Z) The definition of the first difference is: (1-Z)

x(t) = x(t) –x(t-1)x(t) = x(t) –x(t-1) But if we know x(t-1) at time t-1, and we have a But if we know x(t-1) at time t-1, and we have a

forecast for (1-Z) x(t), then we can rearrange the forecast for (1-Z) x(t), then we can rearrange the differencing equation and do summation to differencing equation and do summation to calculate x(t): x(t) = xcalculate x(t): x(t) = x00(t-1) + E(t-1) + Et-1 t-1 (1-Z) x(t)(1-Z) x(t)

This process can be executed on Eviews by using This process can be executed on Eviews by using the Generate commandthe Generate command

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Recoloring In the case of airline passengers, it is easier to In the case of airline passengers, it is easier to

undo the first difference first and then undo the undo the first difference first and then undo the seasonal difference. For this purpose, it is easier to seasonal difference. For this purpose, it is easier to take the transformations in the order, natural log, take the transformations in the order, natural log, seasonal difference, first differenceseasonal difference, first difference

Note: (1-Z)(1-ZNote: (1-Z)(1-Z1212)lnBJPASS(t) = (1-Z)lnBJPASS(t) = (1-Z1212))(1-Z)lnBJPASS(t), I.e the ordering of differencing (1-Z)lnBJPASS(t), I.e the ordering of differencing does not matter does not matter

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SDLNBJPASS

Seasonal difference in thenatural log of airline passengers

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Correlogram of Seasonal Difference in log of passengers.Note there is still structure, decay in the ACF, requiring A first difference to further prewhiten

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DSDLNBJP

First Difference in the Seasonal Difference of theNatural Logarithm of Airline Passengers

As advertised, either order of differencing results in theSame pre-whitened variable

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Using Eviews to Recolor DSDlnBJP(t) = SDlnBJPASS(t) – SDBJPASS(t-1)DSDlnBJP(t) = SDlnBJPASS(t) – SDBJPASS(t-1) DSDlnBJP(1961.01) = SDlnBJPASS(1961.01) – DSDlnBJP(1961.01) = SDlnBJPASS(1961.01) –

SDlnBJPASS(1960.12)SDlnBJPASS(1960.12) So we can rearrange to calculate forecast values of SDlnBJPASS So we can rearrange to calculate forecast values of SDlnBJPASS

from the forecasts for DSDlnBJPfrom the forecasts for DSDlnBJP SDlnBJPASSF(1961.01) = DSDlnBJPF(1961.01) + SDlnBJPASSF(1961.01) = DSDlnBJPF(1961.01) +

SDlnBJPASS(1960.12)SDlnBJPASS(1960.12) We can use this formula in iterative fashion as We can use this formula in iterative fashion as

SDlnBJPASSF(1961.01) = DSDlnBJPF(1961.01) + SDlnBJPASSF(1961.01) = DSDlnBJPF(1961.01) + SDlnBJPASSF(1960.12), but we need an initial value for SDlnBJPASSF(1960.12), but we need an initial value for SDlnBJPASSF(1960.12) since this is the last time period before SDlnBJPASSF(1960.12) since this is the last time period before forecasting. forecasting.

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The initial value

This problem is easily solved by generating This problem is easily solved by generating SDlnBJPASSF(1960.12) = SDlnBJPASSF(1960.12) = SDlnBJPASS(1960.12) SDlnBJPASS(1960.12)

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Recoloring: Generating the forecast of the seasonal difference in lnBJPASS

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SDLNBJPASS SDLNBJPASSF

Forecast of the Seasonal difference in theNatural Logarithm of Airline Passengers

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Recoloring to Undo the Seasonal Difference in the Log of Passengers

Use the definition: SDlnBJPASS(t) = lnBJPASS(t) Use the definition: SDlnBJPASS(t) = lnBJPASS(t) – lnBJPASS(t-12),– lnBJPASS(t-12),

Rearranging and putting in terms of the forecasts Rearranging and putting in terms of the forecasts lnBJPASSF(1961.01) = lnBJPASS(1960.12) + lnBJPASSF(1961.01) = lnBJPASS(1960.12) + SDlnBJPASSF(1961.01)SDlnBJPASSF(1961.01)

In this case we do not need to worry about initial In this case we do not need to worry about initial values in the iteration because we are going back values in the iteration because we are going back twelve months and adding the forecast for the twelve months and adding the forecast for the seasonal differenceseasonal difference

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LNBJPASS LNBJPASSF

Forecast in the Natural Logarithm of Airline Passengers

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The Harder Part is Over

Once the difference and the seasonal Once the difference and the seasonal difference have been undone by summation, difference have been undone by summation, the rest requires less attention to detail, plus the rest requires less attention to detail, plus double checking, to make sure your double checking, to make sure your commands to Eviews were correct.commands to Eviews were correct.

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The Last Step

To convert the forecast of lnBJPASS to the To convert the forecast of lnBJPASS to the forecast of BJPASS use the inverse of the forecast of BJPASS use the inverse of the logarithmic transformation, namely the logarithmic transformation, namely the exponentialexponential

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Twelve Month forward Forecast of Airline Passengers

lab five postscript

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