Identifying ARIMA Models
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1 Identifying ARIMA Identifying ARIMA Models Models What you need to know What you need to know
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Identifying ARIMA Models. What you need to know. Autoregressive of the second order. X(t) = b 1 x(t-1) + b 2 x(t-2) + wn(t) b 2 is the partial regression coefficient measuring the effect of x(t-2) on x(t) holding x(t-1) constant - PowerPoint PPT Presentation
Transcript of Identifying ARIMA Models
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Identifying ARIMA ModelsIdentifying ARIMA Models
What you need to knowWhat you need to know
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Autoregressive of the second orderAutoregressive of the second order
• X(t) = b1 x(t-1) + b2 x(t-2) + wn(t)
• b2 is the partial regression coefficient measuring the effect of x(t-2) on x(t) holding x(t-1) constant
• Since x(t) is regressed on itself lagged, b2
can also be interpreted as a partial autoregression coefficient of x(t) regressed on itself lagged twice.
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continuedcontinued• In one more step b2 can be defined as the
partial autocorrelation coefficient at lag 2, b2 = pacf(2)
• Solving the yule-Walker equations:
• b2 = {acf(2) – [acf(1)]2 }/[1 – [acf(1)]2
• We know that if the process is autoregressive of the first order, then acf(2) = [acf(1)]2 and so b2 = 0
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So now we are back to So now we are back to autoregressive of the first oderautoregressive of the first oder
• x(t) = b x(t-1) + wn(t)
• There is only one regression coefficient, b, so acf(1) = pacf(1) = b
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In summaryIn summary• The partial autocorrelation function, pacf(u)
indicates the order of the autregressive process. If only pacf(1) is significantly different from zero, then the autoregressive process is of order one. If the pacf(2) is significantly different from zero, then the autoregressive process is of order two, and so on.
• Thus we use the partial autocorrelation function to specify the order of the autoregressive process to be estimated
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The autocorrelation functionThe autocorrelation function
• The autocorrelation function, acf(u) is used to determine the order of the moving average process
• If acf(1) is significantly different from zero and there are no other significant autocorrelations, then we specify a first order MA process to be estimated
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Cont.Cont.
• If there is a significant autocorrelation at lag two and none at higher lags, then we specify a second order moving average process
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Moving Average ProcessMoving Average Process
• X(t) = wn(t) + a1wn(t-1) + a2wn(t-2) + a3wn(t-3)
• Taking expectations the mean function is zero, Ex(t) = m(t) = o
• Multiplying by x(t-1) and taking expectations, E[x(t)x(t-1)] =
• EX(t) = wn(t) + a1wn(t-1) + a2wn(t-2) + a3wn(t-3)
X(t-1) = wn(t-1) + a1wn(t-2) + a2wn(t-3) + a3wn(t-4), γx,x (1) = [a1 + a1 a2 + a2 a3 ] σ2
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ContinuingContinuing• The autocovariance at lag 2, γx,x (2) = E x(t) x(t-2)• EX(t) = wn(t) + a1wn(t-1) + a2wn(t-2) + a3wn(t-3)
X(t-2) = wn(t-2) + a1wn(t-3) + a2wn(t-4) + a3wn(t-5), γx,x (2) = [a2 + a1 a3 ] σ2
• The autocovariance at lag 3, γx,x (3) = E x(t) x(t-4)• EX(t) = wn(t) + a1wn(t-1) + a2wn(t-2) + a3wn(t-3)
X(t-3) = wn(t-3) + a1wn(t-4) + a2wn(t-5) + a3wn(t-6), γx,x (3) = [a3 ] σ2
• The autocovariance at lag 4 is zero, E x(t)x(t-4) = 0, so the autocovariance function determines the order of the MA process
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Specifying ARMA ProcessesSpecifying ARMA Processes• x(t) = A(z)/B(z)• The autocovariance function divided by the variance,
i.e. the autocorrelation function, acf(u), indicates the order of A(z) and the partial autocorrelation function, pacf(u) indicates the order of B(z)
• In Eviews specify x(t) c ar(1) ar(2) ….ar(u) for a u th order B(z) and include ma(1) ma(2) ….ma(u) for a uth order A(z),
• i.e. X(t) c ar(1) ar(2) …ar(u) ma(1) ma(2) …ma(u)
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Summary of IdentificationSummary of Identification• Spreadsheet
• Trace: Is it stationary?
• Histogram: is it normal?
• Correlogram: order of A(z) and B(z)
• Unit root test: is it stationary?1111
• Specification
• estimation
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ARMA ProcessesARMA Processes• Identification• Specification and Estimation• Validation
– Significance of estimated parameters and DW– Actual, fitted and residual– Residual tests
• Correlogram: are they orthogonal? Also the Breusch-Godfrey test for serial correlation
• Histogram; are they normal?
• Forecasting
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Example: Capacity utilization mfg.Example: Capacity utilization mfg.
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SpreadsheetSpreadsheet
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HistogramHistogram
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Series: MCUMFNSample 1972:01 2010:03Observations 459
Mean 78.98322Median 79.30000Maximum 88.50000Minimum 65.20000Std. Dev. 4.647918Skewness -0.603442Kurtosis 3.051589
Jarque-Bera 27.90780Probability 0.000001
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CorrelogramCorrelogram
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Unit root testUnit root test
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Pre-WhitenPre-Whiten
Gen dmcumfn =mcumfn – mcumfn(-1)
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SpreadsheetSpreadsheet
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TraceTrace
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DMCUMFN
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histogramhistogram
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Series: DMCUMFNSample 1972:02 2010:03Observations 458
Mean -0.023799Median 0.000000Maximum 1.800000Minimum -3.900000Std. Dev. 0.660698Skewness -1.109196Kurtosis 7.351042
Jarque-Bera 455.1915Probability 0.000000
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CorrelogramCorrelogram
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Unit root testUnit root test
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SpecificationSpecification
Dmcumfn c ar(1) ar(2)
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Estimation
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ValidationValidation
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Residual Actual Fitted
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Correlogram of the residualsCorrelogram of the residuals
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Breusch-Godfrey Serial correlation testBreusch-Godfrey Serial correlation test
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Re-SpecifyRe-Specify
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EstimationEstimation
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ValidationValidation
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Residual Actual Fitted
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Correlogram of the ResidualsCorrelogram of the Residuals
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Breusch-Godfrey Serial correlation testBreusch-Godfrey Serial correlation test
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Histogram of the residualsHistogram of the residuals
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Series: ResidualsSample 1972:05 2010:03Observations 455
Mean 8.09E-05Median 0.010460Maximum 2.669327Minimum -2.954552Std. Dev. 0.583609Skewness -0.316126Kurtosis 6.875041
Jarque-Bera 292.2558Probability 0.000000
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Forecasting: Procs. Workfile rangeForecasting: Procs. Workfile range
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Forecasting: Equation Forecasting: Equation window.forecastwindow.forecast
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ForecastingForecasting
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DMCUMFNF ± 2 S.E.
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Forecasting: Quick, showForecasting: Quick, show
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ForecastingForecasting
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Forecasting: show, view, graph-lineForecasting: show, view, graph-line
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DMCUMFNFORECAST
+2*SEFFORECAST-2*SEF
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ReintegrationReintegration
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Forecasting mcumfnForecasting mcumfn
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Forecast mcumfn, quick, showForecast mcumfn, quick, show
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Forecasting mcumfnForecasting mcumfn
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MCUMFNMCUMFNF
MCUMFNF+2*SEFMCUMFNF-2*SEF
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What can we learn from this forecast?What can we learn from this forecast?• If, in the next nine months, mcumfn grows
beyond the upper bound, this is new information indicating a rebound in manufacturing
• If, in the next nine months, mcumfn stays within the upper and lower bounds, then this means the recovery remains sluggish
• If mcumfn goes below the lower bound, run for the hills!
