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ECON 240CECON 240C

Lecture 10Lecture 10

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2OutlineOutline Box-Jenkins PassengersBox-Jenkins Passengers

Displaying the ForecastDisplaying the Forecast RecoloringRecoloring

ARTWO’s and cyclesARTWO’s and cycles Time seriesTime series Autocorrelation functionAutocorrelation function

Private housing Starts- Single UnitsPrivate housing Starts- Single Units Review: Unit RootsReview: Unit Roots Midterm 2002Midterm 2002

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3Forecasting Seasonal Forecasting Seasonal

Difference in the Difference in the Fractional ChangeFractional Change

Estimation period: 1949.01 – Estimation period: 1949.01 – 1960.121960.12

Forecast period: 1961.01 – 1961.12Forecast period: 1961.01 – 1961.12

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4Eviews forecast command window

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5Eviews plot of forecast plus or minus two standard errors Of the forecast

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6Eviews spreadsheet view of the forecast and the standard Error of the forecast

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7Using 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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9Note: EViews sets the forecast variable equal to the observedValue for 1949.01-1960.12.

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10To Differentiate the To Differentiate the Forecast from the Forecast from the

observed variable ….observed variable …. In the spread sheet window, click In the spread sheet window, click

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

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11Note: EViews sets the forecast variable equal to the observedValue for 1949.01-1960.12.

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12Displaying the ForecastDisplaying the Forecast

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

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13Qick 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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15RecoloringRecoloring

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

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RecoloringRecoloring Summation or integration is the opposite of Summation or integration is the opposite of

differencing.differencing. The definition of the first difference is: The definition of the first difference is:

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

have a forecast for (1-Z) x(t), then we can have a forecast for (1-Z) x(t), then we can rearrange the differencing equation and do rearrange the differencing equation and do summation to calculate x(t): x(t) = xsummation to calculate x(t): x(t) = x00(t-1) + (t-1) + EEt-1 t-1 (1-Z) x(t)(1-Z) x(t)

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

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RecoloringRecoloring In the case of airline passengers, it is In the case of airline passengers, it is

easier to undo the first difference first easier to undo the first difference first and then undo the seasonal difference. and then undo the seasonal difference. For this purpose, it is easier to take the For this purpose, it is easier to take the transformations in the order, natural transformations in the order, natural log, seasonal difference, first differencelog, seasonal 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 (1-Z) lnBJPASS(t), i.e the ordering of differencing does not matter differencing does not matter

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SDLNBJPASS

Seasonal difference in thenatural log of airline passengers

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19Correlogram 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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21Using Eviews to RecolorUsing 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 So we can rearrange to calculate forecast

values of SDlnBJPASS from the forecasts for values of SDlnBJPASS from the forecasts for DSDlnBJPDSDlnBJP

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

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22The initial valueThe initial value

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

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23Recoloring: Generating the Recoloring: Generating the forecast of the seasonal forecast of the seasonal difference in lnBJPASSdifference in lnBJPASS

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SDLNBJPASS SDLNBJPASSF

Forecast of the Seasonal difference in theNatural Logarithm of Airline Passengers

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SDLNBJPAFSDLNBJPAF+2*SEFSDLNBJPAF-2*SEF

Forecast of sdlnbjpaForecast of sdlnbjpa

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26Recoloring to Undo the Recoloring to Undo the

Seasonal Difference in the Seasonal Difference in the Log of PassengersLog of Passengers

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

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

In this case we do not need to worry In this case we do not need to worry about initial values in the iteration about initial values in the iteration because we are going back twelve because we are going back twelve months and adding the forecast for the 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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29lnbjpassflnbjpassf

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LNBJPASSFLNBJPASSF+2*SEFLNBJPASSF-2*SEF

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30The Harder Part is OverThe Harder Part is Over

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

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32The Last StepThe Last Step

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

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BJPASS BJPASSF

Twelve Month forward Forecast of Airline Passengers

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35Confidence IntervalsConfidence Intervals

The confidence interval can be The confidence interval can be generated as:generated as: Lnupper = lnbjpass + 2*sefLnupper = lnbjpass + 2*sef Lnlower = lnbjpassf-2*sefLnlower = lnbjpassf-2*sef And then exponentiated:And then exponentiated: Upper= exp(lnupper)Upper= exp(lnupper) Lower=exp(lnlower)Lower=exp(lnlower)

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37Part I: ARTWO’s and Part I: ARTWO’s and CyclesCycles

ARTWO(t) = bARTWO(t) = b1 1 ARTWO(t-1) + bARTWO(t-1) + b2 2 ARTWO(t-2) ARTWO(t-2) + + WN(t)WN(t)

ARTWO(t) = bARTWO(t) = b1 1 ARTWO(t-1) + bARTWO(t-1) + b2 2 ARTWO(t-2) ARTWO(t-2) is the homogenous deterministic part of is the homogenous deterministic part of the equation after dropping the stochastic the equation after dropping the stochastic part, WN(t).part, WN(t).

Substitute ySubstitute y2-u 2-u for ARTWO(t-u) to obtain: for ARTWO(t-u) to obtain: yy2 2 =b=b1 1 yy11 + b + b2 2 yy00

Or yOr y2 2 - b- b1 1 yy11 - b - b2 2 = 0, which is a quadratic= 0, which is a quadratic

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Note that the corresponding equation Note that the corresponding equation for the autocorrelation function has the for the autocorrelation function has the same behavior:same behavior:

(2) = b(2) = b1 1 (1) + b(1) + b2 2 (0)(0) Let yLet y2-u 2-u 2-u), then2-u), then yy2 2 =b=b1 1 yy11 + b + b2 2 yy00 The same homogeneous equation for The same homogeneous equation for

the autocorrelation function as for the the autocorrelation function as for the process, so if the process cycles so will process, so if the process cycles so will the autocorrelation functionthe autocorrelation function

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39Simulated ARTWOSimulated ARTWO

ARTWO = 0.6*ARTWO(-1) - 0.8*ARTWO(-ARTWO = 0.6*ARTWO(-1) - 0.8*ARTWO(-2)+ WN2)+ WN

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ARTWO

Artwo = 0.6*artwo(-1) - 0.8*artwo(-2) + wn

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42Estimated Coefficients: Simulated ARTWO

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43Privately Owned Housing Privately Owned Housing StartsStarts

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53Part II: Unit RootsPart II: Unit Roots

First Order Autoregressive or First Order Autoregressive or RandomWalk?RandomWalk? y(t) = b*y(t-1) + wn(t)y(t) = b*y(t-1) + wn(t) y(t) = y(t-1) + wn(t)y(t) = y(t-1) + wn(t)

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54Unit RootsUnit Roots

y(t) = b*y(t-1) + wn(t)y(t) = b*y(t-1) + wn(t) we could test the null: b=1 against b<1we could test the null: b=1 against b<1 instead, subtract y(t-1) from both sides:instead, subtract y(t-1) from both sides: y(t) - y(t-1) = b*y(t-1) - y(t-1) + wn(t)y(t) - y(t-1) = b*y(t-1) - y(t-1) + wn(t) or or y(t) = (b -1)*y(t-1) + wn(t) y(t) = (b -1)*y(t-1) + wn(t) so we could regress so we could regress y(t) on y(t-1) and y(t) on y(t-1) and

test the coefficient for y(t-1), i.e.test the coefficient for y(t-1), i.e. y(t) = g*y(t-1) + wn(t), where g = (b-y(t) = g*y(t-1) + wn(t), where g = (b-

1)1) test null: (b-1) = 0 against (b-1)< 0test null: (b-1) = 0 against (b-1)< 0

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55Unit RootsUnit Roots

i.e. test b=1 against b<1i.e. test b=1 against b<1 This would be a simple t-test except This would be a simple t-test except

for a problem. As b gets closer to one, for a problem. As b gets closer to one, the distribution of (b-1) is no longer the distribution of (b-1) is no longer distributed as Student’s t distributiondistributed as Student’s t distribution

Dickey and Fuller simulated many Dickey and Fuller simulated many time series with b=0.99, for example, time series with b=0.99, for example, and looked at the distribution of the and looked at the distribution of the estimated coefficient estimated coefficient

g

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56Unit RootsUnit Roots

Dickey and Fuller tabulated these Dickey and Fuller tabulated these simulated results into Tablessimulated results into Tables

In specifying Dickey-Fuller tests In specifying Dickey-Fuller tests there are three formats: no constant-there are three formats: no constant-no trend, constant-no trend, and no trend, constant-no trend, and constant-trend, and three sets of constant-trend, and three sets of tables.tables.

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57Unit RootsUnit Roots

Example: the price of gold, weekly Example: the price of gold, weekly data, January 1992 through data, January 1992 through December 1999December 1999

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Trace of the Weekly Closing Price of Gold

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Series: GOLDSample 1/06/1992 12/27/1999Observations 417

Mean 345.7320Median 352.5000Maximum 414.5000Minimum 253.8000Std. Dev. 41.31026Skewness -0.514916Kurtosis 1.999475

Jarque-Bera 35.82037Probability 0.000000

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Simulated Student's t-Distribution, 415 Degrees of Freedom

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65Dickey-Fuller TestsDickey-Fuller Tests

The price of gold might vary around a The price of gold might vary around a “constant”, for example the marginal cost “constant”, for example the marginal cost of productionof production

PPGG(t) = MC(t) = MCG G + RW(t) = MC+ RW(t) = MCG G + WN(t)/[1-Z]+ WN(t)/[1-Z]

PPGG(t) - MC(t) - MCG G = RW(t) = WN(t)/[1-Z]= RW(t) = WN(t)/[1-Z]

[1-Z][P[1-Z][PGG(t) - MC(t) - MCG G ] = WN(t)] = WN(t)

[P[PGG(t) - MC(t) - MCG G ] - [P] - [PGG(t-1) - MC(t-1) - MCG G ] = WN(t)] = WN(t)

[P[PGG(t) - MC(t) - MCG G ] = [P] = [PGG(t-1) - MC(t-1) - MCG G ] + WN(t)] + WN(t)

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66Dickey-Fuller TestsDickey-Fuller Tests

Or: [POr: [PGG(t) - MC(t) - MCG G ] = b* [P] = b* [PGG(t-1) - MC(t-1) - MCG G ] + WN(t)] + WN(t)

PPGG(t) = MC(t) = MCG G + b* P + b* PGG(t-1) - b*MC(t-1) - b*MCG G + WN(t) + WN(t)

PPGG(t) = (1-b)*MC(t) = (1-b)*MCG G + b* P + b* PGG(t-1) (t-1) + WN(t) + WN(t)

subtract Psubtract PGG(t-1) (t-1)

PPGG(t) - P(t) - PGG(t-1) = (1-b)*MC(t-1) = (1-b)*MCG G + b* P + b* PGG(t-1) (t-1) - P - PGG(t-(t-1) + WN(t)1) + WN(t)

or or PPGG(t) = (1-b)*MC(t) = (1-b)*MCG G + (1-b)* P + (1-b)* PGG(t-1) + WN(t)(t-1) + WN(t) Now there is an intercept as well as a slopeNow there is an intercept as well as a slope

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69Augmented Dickey- Augmented Dickey- Fuller TestsFuller Tests

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70ARTWO’s and Unit RootsARTWO’s and Unit Roots Recall the edge of the triangle of stability: bRecall the edge of the triangle of stability: b2 2 = 1 – = 1 –

bb1 1 , so for stability b, so for stability b1 1 + b+ b2 2 < 1< 1 x(t) = bx(t) = b1 1 x(t-1) + bx(t-1) + b2 2 x(t-2) + wn(t)x(t-2) + wn(t) Subtract x(t-1) from both sidesSubtract x(t-1) from both sides x(t) – x(-1) = (bx(t) – x(-1) = (b1 1 – 1)x(t-1) + b– 1)x(t-1) + b2 2 x(t-2) + wn(t)x(t-2) + wn(t) Add and subtract bAdd and subtract b2 2 x(t-1) from the right side: x(t-1) from the right side: x(t) – x(-1) = (bx(t) – x(-1) = (b1 1 + b+ b2 2 - 1) x(t-1) - b- 1) x(t-1) - b2 2 [x(t-1) - x(t-2)] + [x(t-1) - x(t-2)] +

wn(t)wn(t) Null hypothesis: (bNull hypothesis: (b1 1 + b+ b2 2 - 1) = 0- 1) = 0 Alternative hypothesis: (bAlternative hypothesis: (b1 1 + b+ b2 2 -1)<0-1)<0

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71Part III. Spring 2002 Part III. Spring 2002 MidtermMidterm

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HELPADS

Figure 4.1 T race of Index of Help Wanted Advertis ing in Newspapers

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Series: HELPADSSample 1951:01 2002:03Observations 615

Mean 66.00325Median 67.00000Maximum 106.0000Minimum 24.00000Std. Dev. 21.92956Skewness -0.080022Kurtosis 1.845852

Jarque-Bera 34.79036Probability 0.000000

Figure 4.2 Histogram/Moments of Index of Help Wanted Advertising

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80Figure 4.4 Dickey-Fuller Unit Root Test for Index of Help Wanted Advertising in Newspapers

ADF Test Statistic -1.354785 1% Critical Value* -3.4434 5% Critical Value -2.8666 10% Critical Value -2.5695

*MacKinnon critical values for rejection of hypothesis of a unit root.

Augmented Dickey-Fuller Test EquationDependent Variable: D(HELPADS)Method: Least Squares

Sample(adjusted): 1951:02 2002:03Included observations: 614 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

HELPADS(-1) -0.005467 0.004036 -1.354785 0.1760C 0.370819 0.280790 1.320631 0.1871

R-squared 0.002990 Mean dependent var 0.009772Adjusted R-squared 0.001361 S.D. dependent var 2.192952S.E. of regression 2.191459 Akaike info criterion 4.410265Sum squared resid 2939.127 Schwarz criterion 4.424662Log likelihood -1351.951 F-statistic 1.835442Durbin-Watson stat 1.929455 Prob(F-statistic) 0.175986

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DLNHELP

Figure 5.1 Trace of Fractional Change in Index of Help Wanted Ads, dlnhelp

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Series: DLNHELPSample 1951:02 2002:03Observations 614

Mean 0.000228Median 0.000000Maximum 0.091567Minimum -0.125880Std. Dev. 0.033695Skewness -0.372063Kurtosis 3.617305

Jarque-Bera 23.91501Probability 0.000006

Figure 5.2 Histogram/Moments of Fractional Change in Index of Help Wanted Ads

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Figure 5.4 Dickey-Fuller Unit Root Test for Fractional Changes in the index of Help Wanted Ads

ADF Test Statistic -20.72793 1% Critical Value* -3.4435 5% Critical Value -2.8666 10% Critical Value -2.5695

*MacKinnon critical values for rejection of hypothesis of a unit root.

Augmented Dickey-Fuller Test EquationDependent Variable: D(DLNHELP)Method: Least Squares

Sample(adjusted): 1951:03 2002:03Included observations: 613 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

DLNHELP(-1) -0.825640 0.039832 -20.72793 0.0000C 0.000142 0.001342 0.105719 0.9158

R-squared 0.412865 Mean dependent var -7.54E-05Adjusted R-squared 0.411904 S.D. dependent var 0.043316S.E. of regression 0.033218 Akaike info criterion -3.968176Sum squared resid 0.674210 Schwarz criterion -3.953761Log likelihood 1218.246 F-statistic 429.6470Durbin-Watson stat 2.113546 Prob(F-statistic) 0.000000

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