f excase problem of multiple regression

7/28/2019 f ExCase Problem of Multiple regression

1/22

s ngBy Assoc. Prof. R Boojhawon

1TSA By Assoc. Prof. R Boojhawon, UoM


2/22



3/22

Building ARIMA Models: Basic Steps

Plotting the data to inspect unusual features like outliers,stationary, seasonal and if variance is stable.

Transforming the data e.g use of log, sqrt, or Box-Coxpower transformation

Identifying the orders (p,d,q) through use of ACF/PACF of

0ln

01

ifx

ifx

y

t

t

t

.may introduce correlation Numerical estimation of model parameters (& CIs) by

using Yule-Walker equations or method of MLE or methodof LSE.

Residual Diagnostics: Graphs of Residual ACF/PACF, QQ,histograms, Ljung-Box-Pierce Q statistics to indicate whitenoise else we need to resimulate using other parameters.

Do predictions

TSA By Assoc. Prof. R Boojhawon, UoM 3


4/22

Goal To emphasize plotting methods that are

appropriate and

useful for finding patterns that will lead to suitable

.



5/22

A Model-Building Strategy

We will develop a multistep model-building strategyespoused so well by Box and Jenkins (1976). There arethree main steps in the process, each of which may be

model specification (or identification)

model fitting, and

model diagnostics

We shall attempt to adhere to the principle ofparsimony



6/22

Case Problem 1The quarterly earnings per share for 19601980 of the U.S. company Johnson &

Johnson, are saved in the file named JJ.

Plot the time series and also the logarithm of the series. Argue that we should

transform by logs to model this series. The series is clearly not stationary. Take first differences and plot that series.

Does stationarity now seem reasonable?

Calculate and graph the sample ACF of the first differences. Interpret theresults.

.plot. Recall that for quarterly data, a season is of length 4.

Graph and interpret the sample ACF of seasonal differences with the firstdifferences.

Fit the model ARIMA(0,1,1)(0,1,1)4, and assess the significance of the estimatedcoefficients.

Perform all of the diagnostic tests on the residuals.

Calculate and plot forecasts for the next two years of the series. Be sure to includeforecast limits.



7/22

Step 1: Time Series Plot of Data/log

of Data



8/22

Step 2: The series is clearly not stationary. Take first differences of

the log data and plot that series. Does stationarity now seem

reasonable?

We do not expect stationary series to have less variability in the middle of the series asthis one does but we might entertain a stationary model and see where it leads us.


l l d h h l f


9/22

Step 3: Calculate and graph the sample ACF of

the first differences of ln(data). Interpret the

results.

Strongest autocorrelations are at the seasonal lags of 4, 8, 12, and 16. Clearly, we need toaddress the seasonality in this series. Also graph suggests MA effect as well.



10/22

PACF of the ln(data) suggesting AR

effects as well



11/22

Step 4: Display the plot of seasonal differences and the first

differences. Interpret the plot. Recall that for quarterly data, a

season is of length 4.

The various quarters seem to be quite randomly distributed among high, middle,and low values (e.g we see 4 up, mid,down), so that most of the seasonality isaccounted for in the seasonal difference.



12/22

Step 5: Graph and interpret the sample ACF of seasonal

differences with the first differences.

They only significant autocorrelations are at lags 1 and 7.Lag 4 (the quarterly lag) is nearly significant. (2 out of 20 = 0.1 can beconsidered as non-significant here)



13/22

Step 6: Fit the model ARIMA(0,1,1)(0,1,1)4, and assess

the significance of the estimated coefficients.

Both the seasonal and nonseasonal ma parameters aresignificant in this model since due to small p-values.



14/22

Step 7: Perform all of the diagnostic tests on the

residuals.

No inadequacies with the model. There is little autocorrelation in theresiduals/Independence due to p-values from LjungBox being large (largep-values) and hence cannot reject Ho: error is IID



15/22

Predicted and Residual values

saved


Histogram


16/22

Histogram



17/22

Normal QQ plot



18/22

Normally Test



19/22

Step 8: Calculate and plot forecasts for the next two

years of the series. Be sure to include forecast limits.

Forecasts follow the general pattern of seasonality and trendForecast limits give a good indication of the confidence in these forecasts.



20/22

Returning back to the forecast of

data in original terms



21/22

Predicted and Residual values

saved



22/22

ConclusionWe may extend our models using higher parameters

but it is good practice to use the simplest one (leastparameters: Principle of Parsimony) which satisfies

.


f excase problem of multiple regression

Documents