applied business forecasting and planning multiple regression analysis

Applied Business Forecasting and planning

Multiple Regression Analysis

Introduction In simple linear regression we studied the

relationship between one explanatory variable and one response variable.

Now, we look at situations where several explanatory variables works together to explain the response.

Introduction Following our principles of data analysis,

we look first at each variable separately, then at relationships among the variables.

We look at the distribution of each variable to be used in multiple regression to determine if there are any unusual patterns that may be important in building our regression analysis.

Multiple Regression

Example. In a study of direct operating cost, Y, for 67 branch offices of consumer finance charge, four independent variables were considered:

X1: Average size of loan outstanding during the year,

X2 : Average number of loans outstanding,

X3 : Total number of new loan applications processed, and

X4 : Office salary scale index.

The model for this example is 443322110 xXxxY

Formal Statement of the Model General regression model

0, 1, , k are parameters

• X1, X2, …,Xk are known constants , the error terms are independent N(o, 2)

kk xxxY 22110

Estimating the parameters of the model

The values of the regression parameters i are not known. We estimate them from data.

As in the simple linear regression case, we use the least-squares method to fit a linear function

to the data. The least-squares method chooses the b’s that make

the sum of squares of the residuals as small as possible.

kk xbxbxbby 22110ˆ


The least-squares estimates are the values that minimize the quantity

Since the formulas for the least-squares estimates are complicated and hand calculation is out of question, we are content to understand the least-squares principle and let software do the computations.

2

1

)ˆ(

n

iii yy


The estimate of i is bi and it indicates the change in the mean response per unit increase in Xi when the rest of the independent variables in the model are held constant.

The parameters i are frequently called partial regression coefficients because they reflect the partial effect of one independent variable when the rest of independent variables are included in the model and are held constant


The observed variability of the responses about this fitted model is measured by the variance

and the regression standard error

n

iii yy

knS

1

22 )ˆ(1

1

2ss


In the model 2 and measure the variability of the responses about the population regression equation.

It is natural to estimate 2 by s2 and by s.

Analysis of Variance Table The basic idea of the regression ANOVA table are

the same in simple and multiple regression. The sum of squares decomposition and the

associated degrees of freedom are:

df:SSESSRSST

yyyyyy iiii

222 )ˆ()ˆ()(

)1(1 knkn

Analysis of Variance Table

Source Sum of Squares

df Mean Square

F-test

Regression SSR k MSR=

SSR/k

MSR/MSE

Error SSE n-k-1 MSE=

SSE/n-k-1

Total SST n-1

F-test for the overall fit of the model

To test the statistical significance of the regression relation between the response variable y and the set of variables x1,…, xk, i.e. to choose between the alternatives:

We use the test statistic:zero equal ),1( allnot :

0: 210

kiH

H

ia

k

MSE

MSRF

F-test for the overall fit of the model

The decision rule at significance level is: Reject H0 if

Where the critical value F(, k, n-k-1) can be found from an F-table.

The existence of a regression relation by itself does not assure that useful prediction can be made by using it.

Note that when k=1, this test reduces to the F-test for testing in simple linear regression whether or not 1= 0

)1,;( knkFF

Interval estimation of i

For our regression model, we have:

Therefore, an interval estimate for i

with 1- confidence coefficient is:

Where

freedom of degrees 1-k-non with distributi- ta has)( i

ii

bs

b

)()1;2

( ii bskntb

2)()(

xx

MSEbs i

Significance tests for i

To test:

We may use the test statistic:

Reject H0 if

0:

0:0

ia

i

H

H

)( i

i

bs

bt

)1;2

(

)1;2

(

kntt

orkntt

Multiple regression model Building

Often we have many explanatory variables, and our goal is to use these to explain the variation in the response variable.

A model using just a few of the variables often predicts about as well as the model using all the explanatory variables.


We may find that the reciprocal of a variable is a better choice than the variable itself, or that including the square of an explanatory variable improves prediction.

We may find that the effect of one explanatory variable may depends upon the value of another explanatory variable. We account for this situation by including interaction terms.


The simplest way to construct an interaction term is to multiply the two explanatory variables together.

How can we find a good model?

Selecting the best Regression equation.

After a lengthy list of potentially useful independent variables has been compiled, some of the independent variables can be screened out. An independent variable May not be fundamental to the problem May be subject to large measurement error May effectively duplicate another independent

variable in the list.

Selecting the best Regression Equation.

Once the investigator has tentatively decided upon the functional forms of the regression relations (linear, quadratic, etc.), the next step is to obtain a subset of the explanatory variables (x) that “best” explain the variability in the response variable y.

Selecting the best Regression Equation.

An automatic search procedure that develops sequentially the subset of explanatory variables to be included in the regression model is called stepwise procedure.

It was developed to economize on computational efforts.

It will end with the identification of a single regression model as “best”.

Example: Sales Forecasting Sales Forecasting

Multiple regression is a popular technique for predicting product sales with the help of other variables that are likely to have a bearing on sales.

Example The growth of cable television has created vast new potential

in the home entertainment business. The following table gives the values of several variables measured in a random sample of 20 local television stations which offer their programming to cable subscribers. A TV industry analyst wants to build a statistical model for predicting the number of subscribers that a cable station can expect.

Example:Sales Forecasting Y = Number of cable subscribers (SUSCRIB) X1 = Advertising rate which the station charges local

advertisers for one minute of prim time space (ADRATE)

X2 = Kilowatt power of the station’s non-cable signal (KILOWATT)

X3 = Number of families living in the station’s area of dominant influence (ADI), a geographical

division of radio and TV audiences (APIPOP) X4 = Number of competing stations in the ADI

(COMPETE)

Example:Sales Forecasting

The sample data are fitted by a multiple regression model using Excel program.

The marginal t-test provides a way of choosing the variables for inclusion in the equation.

The fitted Model is

SIGNALCOMPETE 43210 APIPOPADRATESUBSCRIBE


Excel Summary outputSUMMARY OUTPUT

Regression StatisticsMultiple R 0.884267744R Square 0.781929444Adjusted R Square 0.723777295Standard Error 142.9354188Observations 20

ANOVAdf SS MS F Significance F

Regression 4 1098857.84 274714.4601 13.44626923 7.52E-05Residual 15 306458.0092 20430.53395Total 19 1405315.85

Coefficients Standard Error t Stat P-value Lower 95%Upper 95%Intercept 51.42007002 98.97458277 0.51952803 0.610973806 -159.539 262.3795AD_Rate -0.267196347 0.081055107 -3.296477624 0.004894126 -0.43996 -0.09443Signal -0.020105139 0.045184758 -0.444954014 0.662706578 -0.11641 0.076204APIPOP 0.440333955 0.135200486 3.256896248 0.005307766 0.152161 0.728507Compete 16.230071 26.47854322 0.61295181 0.549089662 -40.2076 72.66778


Do we need all the four variables in the model?

Based on the partial t-test, the variables signal and compete are the least significant variables in our model.

Let’s drop the least significant variables one at a time.


Excel Summary OutputSUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 51.31610447 96.4618242 0.531983558 0.602046756 -153.1737817 255.806AD_Rate -0.259538026 0.077195983 -3.36206646 0.003965102 -0.423186162 -0.09589APIPOP 0.433505145 0.130916687 3.311305499 0.004412929 0.15597423 0.711036Compete 13.92154404 25.30614013 0.550125146 0.589831583 -39.72506442 67.56815


The variable Compete is the next variable to get rid of.


Excel Summary OutputSUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 96.28121395 50.16415506 1.919322948 0.07188916 -9.556049653 202.1184776AD_Rate -0.254280696 0.075014548 -3.389751739 0.003484198 -0.41254778 -0.096013612APIPOP 0.495481252 0.065306012 7.587069489 7.45293E-07 0.357697418 0.633265086


• All the variables in the model are statistically significant, therefore our final model is:

Final Model

APIPOP495.0ADRATE25.028.96SUBSCRIBE

Interpreting the Final Model

What is the interpretation of the estimated parameters. Is the association positive or negative? Does this make sense intuitively, based on what the data

represents? What other variables could be confounders? Are there other analysis that you might consider doing?

New questions raised?

Multicollinearity In multiple regression analysis, one is often

concerned with the nature and significance of the relations between the explanatory variables and the response variable.

Questions that are frequently asked are: What is the relative importance of the effects of the

different independent variables? What is the magnitude of the effect of a given

independent variable on the dependent variable?

Multicollinearity Can any independent variable be dropped from the

model because it has little or no effect on the dependent variable?

Should any independent variables not yet included in the model be considered for possible inclusion?

Simple answers can be given to these questions if The independent variables in the model are

uncorrelated among themselves. They are uncorrelated with any other independent

variables that are related to the dependent variable but omitted from the model.

Multicollinearity When the independent variables are correlated among

themselves, multicollinearity or colinearity among them is said to exist.

In many non-experimental situations in business, economics, and the social and biological sciences, the independent variables tend to be correlated among themselves.

For example, in a regression of family food expenditures on the variables: family income, family savings, and the age of head of household, the explanatory variables will be correlated among themselves.

Multicollinearity Further, the explanatory variables will also

be correlated with other socioeconomic variables not included in the model that do affect family food expenditures, such as family size.

Multicollinearity Some key problems that typically arise when the

explanatory variables being considered for the regression model are highly correlated among themselves are:

1. Adding or deleting an explanatory variable changes the regression coefficients.

2. The estimated standard deviations of the regression coefficients become large when the explanatory variables in the regression model are highly correlated with each other.

3. The estimated regression coefficients individually may not be statistically significant even though a definite statistical relation exists between the response variable and the set of explanatory variables.

Multicollinearity Diagnostics A formal method of detecting the presence of

multicollinearity that is widely used is by the means of Variance Inflation Factor. It measures how much the variances of the estimated

regression coefficients are inflated as compared to when the independent variables are not linearly related.

Is the coefficient of determination from the regression of the jth independent variable on the remaining k-1 independent variables.

kjR

VIFj

j ,2,1,1

12

2jR

Multicollinearity Diagnostics AVIF near 1 suggests that multicollinearity is not a

problem for the independent variables. Its estimated coefficient and associated t value will not change

much as the other independent variables are added or deleted from the regression equation.

A VIF much greater than 1 indicates the presence of multicollinearity. A maximum VIF value in excess of 10 is often taken as an indication that the multicollinearity may be unduly influencing the least square estimates. the estimated coefficient attached to the variable is unstable and

its associated t statistic may change considerably as the other independent variables are added or deleted.

Multicollinearity Diagnostics The simple correlation coefficient between all pairs

of explanatory variables (i.e., X1, X2, …, Xk ) is helpful in selecting appropriate explanatory variables for a regression model and is also critical for examining multicollinearity.

While it is true that a correlation very close to +1 or –1 does suggest multicollinearity, it is not true (unless there are only two explanatory variables) to infer multicollinearity does not exist when there are no high correlations between any pair of explanatory variables.

Example:Sales ForecastingPearson Correlation Coefficients, N = 20

Prob > |r| under H0: Rho=0 SUBSCRIB ADRATE KILOWATT APIPOP COMPETE SUBSCRIB 1.00000 -0.02848 0.44762 0.90447 0.79832 SUBSCRIB 0.9051 0.0478 <.0001 <.0001 ADRATE -0.02848 1.00000 -0.01021 0.32512 0.34147 ADRATE 0.9051 0.9659 0.1619 0.1406 KILOWATT 0.44762 -0.01021 1.00000 0.45303 0.46895 KILOWATT 0.0478 0.9659 0.0449 0.0370 APIPOP 0.90447 0.32512 0.45303 1.00000 0.87592 APIPOP <.0001 0.1619 0.0449 <.0001 COMPETE 0.79832 0.34147 0.46895 0.87592 1.00000 COMPETE <.0001 0.1406 0.0370 <.0001


APIPOP495.0ADRATE25.028.96SUBSCRIBE

COMPETE16.23APIPOP44.0SIGNAL.02-ADRATE27.042.51SUBSCRIBE

COMPETE13.92APIPOP43.0ADRATE26.032.51SUBSCRIBE


VIF calculation: Fit the model

COMPETE 3210 ADRATESIGNALAPIPOP SUMMARY OUTPUT



Regression 3 3762601 1254200 17.9541 2.25472E-05Residual 16 1117695 69855.92Total 19 4880295

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept -472.685 139.7492 -3.38238 0.003799 -768.9402258 -176.43Compete 159.8413 28.29157 5.649786 3.62E-05 99.86587622 219.8168ADRATE 0.048173 0.149395 0.322455 0.751283 -0.268529713 0.364876Signal 0.037937 0.083011 0.457012 0.653806 -0.138038952 0.213913


Fit the modelSIGNAL 3210 APIPOPADRATECompete

SUMMARY OUTPUT




CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept 3.10416 0.520589 5.96278 1.99E-05 2.000559786 4.20776ADRATE 0.000491 0.000755 0.649331 0.525337 -0.001110874 0.002092Signal 0.000334 0.000418 0.799258 0.435846 -0.000552489 0.001221APIPOP 0.004167 0.000738 5.649786 3.62E-05 0.002603667 0.005731


Fit the modelCOMPETE 3210 APIPOPADRATESignal

SUMMARY OUTPUT



Regression 3 3559789 1186596 1.897261 0.170774675Residual 16 10006813 625425.8Total 19 13566602

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept 5.171093 547.6089 0.009443 0.992582 -1155.707711 1166.05APIPOP 0.339655 0.743207 0.457012 0.653806 -1.235874129 1.915184Compete 114.8227 143.6617 0.799258 0.435846 -189.7263711 419.3718ADRATE -0.38091 0.438238 -0.86919 0.397593 -1.309935875 0.548109


Fit the modelCOMPETE 3210 APIPOPSignalADRATE

SUMMARY OUTPUT



Regression 3 589101.7 196367.2 1.010346 0.413876018Residual 16 3109703 194356.5Total 19 3698805

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept 253.7304 298.6063 0.849716 0.408018 -379.2865355 886.7474Signal -0.11837 0.136186 -0.86919 0.397593 -0.407073832 0.170329APIPOP 0.134029 0.415653 0.322455 0.751283 -0.747116077 1.015175Compete 52.3446 80.61309 0.649331 0.525337 -118.5474784 223.2367


VIF calculation Results:

There is no significant multicollinearity.

Variable R- Squared VIFADRATE 0.159268 1.19COMPETE 0.779575 4.54SIGNAL 0.262394 1.36APIPOP 0.770978 4.36

Qualitative Independent Variables

Many variables of interest in business, economics, and social and biological sciences are not quantitative but are qualitative.

Examples of qualitative variables are gender (male, female), purchase status (purchase, no purchase), and type of firms.

Qualitative variables can also be used in multiple regression.

Qualitative Independent Variables An economist wished to relate the speed with which a

particular insurance innovation is adopted (y) to the size of the insurance firm (x1) and the type of firm. The dependent variable is measured by the number of months elapsed between the time the first firm adopted the innovation and and the time the given firm adopted the innovation. The first independent variable, size of the firm, is quantitative, and measured by the amount of total assets of the firm. The second independent variable, type of firm, is qualitative and is composed of two classes-Stock companies and mutual companies.

Indicator variables Indicator, or dummy variables are used to

determine the relationship between qualitative independent variables and a dependent variable.

Indicator variables take on the values 0 and 1. For the insurance innovation example, where the

qualitative variable has two classes, we might define the indicator variable x2 as follows:

otherwise0

companystock if12 x

Indicator variables A qualitative variable with c classes will be

represented by c-1 indicator variables. A regression function with an indicator

variable with two levels (c = 2) will yield two estimated lines.

Interpretation of Regression Coefficients

In our insurance innovation example, the regression model is:

Where:

22110 xxy

firm of size 1 x

otherwise0

companystock if12 x


To understand the meaning of the regression coefficients in this model, consider first the case of mutual firm. For such a firm, x2 = 0 and we have:

For a stock firm x2 = 1 and the response

function is:

firmsMutualxbbbxbbyi 1102110 )0(ˆ

firmsStockxbbbbxbbyi 11202110 )()1(ˆ


The response function for the mutual firms is a straight line, with y intercept 0 and slope 1.

For stock firms, this also is a straight line, with the same slope 1 but with y intercept 0+2.

With reference to the insurance innovation example, the mean time elapsed before the innovation is adopted is linear function of size of firm (x1), with the same slope 1for both types of firms.


2 indicates how much lower or higher the response function for stock firm is than the one for the mutual firm.

2 measures the differential effect of type of firms.

In general, 2 shows how much higher (lower) the mean response line is for the class coded 1 than the line for the class coded 0, for any level of x1.

Example: Insurance Innovation Adoption

Here is the data set for the insurance innovation example:Months Elapsed Size type of firm Type

17 151 0 Mutual26 92 0 Mutual21 175 0 Mutual30 31 0 Mutual22 104 0 Mutual0 277 0 Mutual12 210 0 Mutual19 120 0 Mutual4 290 0 Mutual16 238 1 Stock28 164 1 Stock15 272 1 Stock11 295 1 Stock38 68 1 Stock31 85 1 Stock21 224 1 Stock20 166 1 Stock13 305 1 Stock30 124 1 Stock14 246 1 Stock


Fitting the regression model

Where

fitted response function is:

22110 xxy

firm of size 1 x

otherwise0

companystock if12 x

21 77.81061.87.33ˆ xxy


SUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 33.8698658 1.562588138 21.67549 8E-14 30.57308841 37.16664321Size -0.10608882 0.007799653 -13.6017 1.45E-10 -0.122544675 -0.089632969type of firm 8.76797549 1.286421264 6.815789 3.01E-06 6.053860079 11.4820909


The fitted response function is:

Stock firms response function is:

Mutual firms response function is:

Interpretation ?

21 77.81061.87.33ˆ xxy

11061.)77.887.33(ˆ xy

11061.87.33ˆ xy

Accounting for Seasonality in a Multiple regression Model

Seasonal Patterns are not easily accounted for by the typical causal variables that we use in regression analysis.

An indicator variable can be used effectively to account for seasonality in our time series data.

The number of seasonal indicator variables to use depends on the data.

If we have p periods in our data series, we can not use more than P-1 seasonal indicator variables.

Example: Private Housing Starts (PHS)

Housing starts in the United States measured in thousands of units. These data are plotted for 1990 Q1 through 1999Q4. There are typically few housing starts during the first quarter of the year (January, February, March); there is usually a big increase in the second quarter of (April, May, June), followed by some decline in the third quarter (July, August, September), and further decline in the fourth quarter (October, November, December).


Private Housing Starts (PHS) in Thousands of Units

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1 1

1

1

1

1 1

1

"1" marks the first quarter of each year.


To Account for and measure this seasonality in a regression model, we will use three dummy variables: Q2 for the second quarter, Q3 for the third quarter, and Q4 for the fourth quarter. These will be coded as follows: Q2 = 1 for all second quarters and zero otherwise. Q3 = 1 for all third quarters and zero otherwise Q4 = 1 for all fourth quarters and zero otherwise.


Data for private housing starts (PHS), the mortgage rate (MR), and these seasonal indicator variables are shown in the following slide.

Examine the data carefully to verify your understanding of the coding for Q2, Q3, Q4.

Since we have assigned dummy variables for the second, third, and fourth quarters, the first quarter is the base quarter for our regression model.

Note that any quarter could be used as the base, with indicator variables to adjust for differences in other quarters.

Example: Private Housing Starts (PHS)PERIOD PHS MR Q2 Q3 Q4

31-Mar-90 217 10.1202 0 0 030-Jun-90 271.3 10.3372 1 0 030-Sep-90 233 10.1033 0 1 031-Dec-90 173.6 9.9547 0 0 131-Mar-91 146.7 9.5008 0 0 030-Jun-91 254.1 9.5265 1 0 030-Sep-91 239.8 9.2755 0 1 031-Dec-91 199.8 8.6882 0 0 131-Mar-92 218.5 8.7098 0 0 030-Jun-92 296.4 8.6782 1 0 030-Sep-92 276.4 8.0085 0 1 031-Dec-92 238.8 8.2052 0 0 131-Mar-93 213.2 7.7332 0 0 030-Jun-93 323.7 7.4515 1 0 030-Sep-93 309.3 7.0778 0 1 031-Dec-93 279.4 7.0537 0 0 131-Mar-94 252.6 7.2958 0 0 030-Jun-94 354.2 8.4370 1 0 030-Sep-94 325.7 8.5882 0 1 031-Dec-94 265.9 9.0977 0 0 131-Mar-95 214.2 8.8123 0 0 030-Jun-95 296.7 7.9470 1 0 030-Sep-95 308.2 7.7012 0 1 031-Dec-95 257.2 7.3508 0 0 131-Mar-96 240 7.2430 0 0 030-Jun-96 344.5 8.1050 1 0 030-Sep-96 324 8.1590 0 1 031-Dec-96 252.4 7.7102 0 0 131-Mar-97 237.8 7.7905 0 0 030-Jun-97 324.5 7.9255 1 0 030-Sep-97 314.6 7.4692 0 1 031-Dec-97 256.8 7.1980 0 0 131-Mar-98 258.4 7.0547 0 0 030-Jun-98 360.4 7.0938 1 0 030-Sep-98 348 6.8657 0 1 031-Dec-98 304.6 6.7633 0 0 131-Mar-99 294.1 6.8805 0 0 030-Jun-99 377.1 7.2037 1 0 030-Sep-99 355.6 7.7990 0 1 031-Dec-99 308.1 7.8338 0 0 1


The regression model for private housing starts (PHS) is:

In this model we expect b1 to have a negative sign, and we would expect b2, b3, b4 all to have positive signs. Why?

Regression results for this model are shown in the next slide.

)4()3()2()( 43210 QQQMRPHS

Example: Private Housing Starts (PHS)SUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 473.0650749 35.54169837 13.31014264 2.93931E-15 400.9115031 545.2186467MR -30.04838192 4.257226391 -7.058206249 3.21421E-08 -38.69102153 -21.40574231Q2 95.74106935 11.84748487 8.081130334 1.6292E-09 71.689367 119.7927717Q3 73.92904763 11.82881519 6.249911462 3.62313E-07 49.91524679 97.94284847Q4 20.54778131 11.84139803 1.73524961 0.091495355 -3.491564078 44.5871267


Use the prediction equation to make a forecast for each of the fourth quarter of 1999.

Prediction equation:)4(55.20)3(93.73)2(74.95)(05.3006.473ˆ QQQMRSPH


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PHS PHSF1 PHSF2

Private Housing Starts (PHS) with a Simple Regression Forecast (PHSF1) and a Multiple Regression Forecast (PHSF2) in Thousands of Units

Regression Diagnostics and Residual Analysis

It is important to check the adequacy of the model before it becomes part of the decision making process.

Residual plots can be used to check the model assumptions.

It is important to study outlying observations to decide whether they should be retained or eliminated.

If retained, whether their influence should be reduced in the fitting process or revise the regression function.

Time Series Data and the Problem of Serial Correlation

In the regression models we assume that the errors i are independent.

In business and economics, many regression applications involve time series data.

For such data, the assumption of uncorrelated or independent error terms is often not appropriate.

Problems of Serial Correlation If the error terms in the regression model are

autocorrelated, the use of ordinary least squares procedures has a number of important consequences MSE underestimate the variance of the error terms The confidence intervals and tests using the t and F

distribution are no longer strictly applicable. The standard error of the regression coefficients

underestimate the variability of the estimated regression coefficients. Spurious regression can result.

First order serial correlation The error term in current period is directly related to the error

term in the previous time period. Let the subscript t represent time, then the simple linear

regression model is:

Where

t = error at time t = the parameter that measures correlation between adjacent error

terms t normally distributed error terms with mean zero and variance 2

ttt xy 10

ttt 1

Example The effect of positive serial correlation in a

simple linear regression model. Misleading forecasts of future y values. Standard error of the estimate, S y.x will

underestimate the variability of the y’s about the true regression line.

Strong autocorrelation can make two unrelated variables appear to be related.

Durbin-Watson Test for Serial Correlation Recall the first-order serial correlation model

The hypothesis to be tested are:

The alternative hypothesis is > 0 since in business and economic time series tend to show positive correlation.

ttt 1

0:

0:0

aH

H

ttt xy 10

Durbin-Watson Test for Serial Correlation

The Durbin-Watson statistic is defined as

Where

n

tt

n

ttt

e

eeDW

1

2

2

21)(

tperiod for time residual theˆ ttt yye

1- tperiod for time residual theˆ 111 ttt yye


The auto correlation coefficient can be estimated by the lag 1 residual autocorrelation r1(e)

And it can be shown that

n

tt

n

ttt

e

eeer

1

2

21

1 )(

))(1(2 1 erDW


Since –1 < r1(e) < 1 then 0 < DW < 4

If r1(e) = 0, then DW = 2 (there is no correlation.)

If r1(e) > 0, then DW < 2 (positive correlation)

If r1(e) < 0, Then DW > 2 (negative correlation)

Durbin-Watson Test for Serial Correlation Decision rule:

If DW > U, Do not reject H0. If DW < L, Reject H0

If L DW U, the test is inconclusive. The critical Upper (U) an Lower (L) bound can be

found in Durbin-Watson table of your text book. To use this table you need to know The

significance level () The number of independent parameters in the model (k), and the sample size (n).

Example The Blaisdell Company wished to predict

its sales by using industry sales as a predictor variable. The following table gives seasonally adjusted quarterly data on company sales and industry sales for the period 1983-1987.

ExampleYear Quarter t CompSale InduSale1983 1 1 20.96 127.3

2 2 21.4 1303 3 21.96 132.74 4 21.52 129.4

1984 1 5 22.39 1352 6 22.76 137.13 7 23.48 141.24 8 23.66 142.8

1985 1 9 24.1 145.52 10 24.01 145.33 11 24.54 148.34 12 24.3 146.4

1986 1 13 25 150.22 14 25.64 153.13 15 26.36 157.34 16 26.98 160.7

1987 1 17 27.52 164.22 18 27.78 165.63 19 28.24 168.74 20 28.78 171.7

Example

Blaisdell Company Example

0

5

10

15

20

25

30

35

0 50 100 150 200

Industry sales($ millions)

Co

mp

any

Sal

es (

$ m

illi

on

s)

Example The scatter plot suggests that a linear regression

model is appropriate. Least squares method was used to fit a regression

line to the data. The residuals were plotted against the fitted

values. The plot shows that the residuals are consistently

above or below the fitted value for extended periods.

Example

Example To confirm this graphic diagnosis we will use the Durbin-

Watson test for:

The test statistic is:0:

0:0

aH

H

n

tt

n

ttt

e

eeDW

1

2

2

21)(

ExampleYear Quarter t Company sales(y) Industry sales(x) et et-et-1 (et-et-1) 2̂ et 2̂

1983 1 1 20.96 127.3 -0.02605 0.0006792 2 21.4 130 -0.06202 -0.03596 0.001293 0.0038463 3 21.96 132.7 0.022021 0.084036 0.007062 0.0004854 4 21.52 129.4 0.163754 0.141733 0.020088 0.026815

1984 1 5 22.39 135 0.04657 -0.11718 0.013732 0.0021692 6 22.76 137.1 0.046377 -0.00019 3.76E-08 0.0021513 7 23.48 141.2 0.043617 -0.00276 7.61E-06 0.0019024 8 23.66 142.8 -0.05844 -0.10205 0.010415 0.003415

1985 1 9 24.1 145.5 -0.0944 -0.03596 0.001293 0.0089112 10 24.01 145.3 -0.14914 -0.05474 0.002997 0.0222433 11 24.54 148.3 -0.14799 0.001152 1.33E-06 0.0219014 12 24.3 146.4 -0.05305 0.094937 0.009013 0.002815

1986 1 13 25 150.2 -0.02293 0.030125 0.000908 0.0005262 14 25.64 153.1 0.105852 0.12878 0.016584 0.0112053 15 26.36 157.3 0.085464 -0.02039 0.000416 0.0073044 16 26.98 160.7 0.106102 0.020638 0.000426 0.011258

1987 1 17 27.52 164.2 0.029112 -0.07699 0.005927 0.0008482 18 27.78 165.6 0.042316 0.013204 0.000174 0.0017913 19 28.24 168.7 -0.04416 -0.08648 0.007478 0.001954 20 28.78 171.7 -0.03301 0.011152 0.000124 0.00109

0.097941 0.133302Blaisdell Company Example

0

5

10

15

20

25

30

35

0 50 100 150 200

Industry sales($ millions)

Co

mp

any

Sal

es (

$ m

illio

ns)

Example

Using Durbin Watson table of your text book, for k = 1, and n=20, and using = .01 we find U = 1.15, and L = .95

Since DW = .735 falls below L = .95 , we reject the null hypothesis, namely, that the error terms are positively autocorrelated.

735.13330.

09794.DW

Remedial Measures for Serial Correlation

Addition of one or more independent variables to the regression model. One major cause of autocorrelated error terms

is the omission from the model of one or more key variables that have time-ordered effects on the dependent variable.

Use transformed variables. The regression model is specified in terms of

changes rather than levels.

Extensions of the Multiple Regression Model

In some situations, nonlinear terms may be needed as independent variables in a regression analysis. Business or economic logic may suggest that non-

linearity is expected. A graphic display of the data may be helpful in

determining whether non-linearity is present. One common economic cause for non-linearity is

diminishing returns. Fore example, the effect of advertising on sales may

diminish as increased advertising is used.


Some common forms of nonlinear functions are :

)()( 2210 XXY

)()()( 33

2210 XXXY

)1(10 XY

10 XeY


To illustrate the use and interpretation of a non-linear term, we return to the problem of developing a forecasting model for private housing starts (PHS).

So far we have looked at the following model

Where MR is the mortgage rate and Q2, Q3, and Q4 are indicators variables for quarters 2, 3, and 4.

)4()3()2()( 43210 QQQMRPHS

Example: Private Housing Start

First we add real disposable personal income per capita (DPI) as an independent variable. Our new model for this data set is:


)()4()3()2()( 543210 DPIQQQMRPHS

Example: Private Housing StartSUMMARY OUTPUT



Regression 5 97690.01942 19538 53.80753 6.51194E-15Residual 33 11982.59955 363.1091Total 38 109672.619

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept -31.06403714 105.1938477 -0.2953 0.769613 -245.0826992 182.9546249MR -20.1992545 4.124906847 -4.8969 2.5E-05 -28.59144723 -11.80706176Q2 97.03478074 8.900711541 10.90191 1.78E-12 78.9261326 115.1434289Q3 75.40017073 8.827185877 8.541813 7.17E-10 57.44111179 93.35922967Q4 20.35306822 8.83373887 2.304015 0.027657 2.380677107 38.32545934DPI 0.022407799 0.004356973 5.142974 1.21E-05 0.013543464 0.031272134


The prediction model is

In comparison with the previous model, we see that the R-squared has improved.It has changed from 78% to 89%.

The standard error of the estimate has decreased from 26.49 for the previous model to 19.05 for the new model.

)(02.0)4(35.20)3(40.75)2(03.97)(19.2006.31ˆ DPIQQQMRSPH


The value of the DW test has changed from 0.88 for the previous model to 0.78 for the new model.

At 5% level the critical value for DW test, from Durbin-Watson table, for k = 5, and n = 39 is L= 1.22, and U = 1.79.

Since The value of the DW test is smaller than L=1.22, we reject the null hypothesis H0: =0

This implies that there is serial correlation in both models, the assumption of the independence of the error terms is not valid.


The Plot of PHS against DPI shows a curve linear relation.

Next we introduce a nonlinear term into the regression.

The square of disposable personal income per capita (DPI2) is included in the regression model.

Private Housing Start and Disposable Personal Income

17500

18000

18500

19000

19500

20000

20500

21000

21500

0 50 100 150 200 250 300 350 400

DPI

PHS


We also add the dependent variable, lagged one quarter, as an independent variable in order to help reduce serial correlation.

The third model that we fit to our data set is:


)()()()4()3()2()( 72

6543210 LPHSDPIDPIQQQMRPHS

Example: Private Housing StartSUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 716.5926532 1017.664989 0.704153784 0.486593 -1358.949934 2792.13524MR -13.65521724 3.093504134 -4.414158396 0.000114 -19.96446404 -7.345970448Q2 106.9813297 6.069780998 17.62523718 1.04E-17 94.60192287 119.3607366Q3 27.72122303 9.111432565 3.042465916 0.004748 9.138323433 46.30412262Q4 -13.37855186 7.653050858 -1.748133144 0.09034 -28.98706069 2.22995698DPI -0.060399279 0.104412354 -0.578468704 0.567127 -0.273349798 0.15255124DPI SQUARED 0.000335974 0.000536397 0.626354647 0.535668 -0.000758014 0.001429963LPHS 0.655786939 0.097265424 6.742241114 1.51E-07 0.457412689 0.854161189


The inclusion of DPI2 and Lagged PHS has increased the R-squared to 96%

The standard error of the estimate has decreased to 12.45

The value of the DW test has increased to 2.32 which is greater than U = 1.79 which rule out positive serial correlation.

You see that the third model worked best for this data set.

The following slide gives the data set.

Example: Private Housing StartPERIOD PHS MR LPHS Q2 Q3 Q4 DPI DPI SQUARED

30-Jun-90 271.3 10.3372 217 1 0 0 18063 1,631,359.85

30-Sep-90 233 10.1033 271.3 0 1 0 18031 1,625,584.81

31-Dec-90 173.6 9.9547 233 0 0 1 17856 1,594,183.68

31-Mar-91 146.7 9.5008 173.6 0 0 0 17748 1,574,957.52

30-Jun-91 254.1 9.5265 146.7 1 0 0 17861 1,595,076.61

30-Sep-91 239.8 9.2755 254.1 0 1 0 17816 1,587,049.28

31-Dec-91 199.8 8.6882 239.8 0 0 1 17811 1,586,158.61

31-Mar-92 218.5 8.7098 199.8 0 0 0 18000 1,620,000.00

30-Jun-92 296.4 8.6782 218.5 1 0 0 18085 1,635,336.13

30-Sep-92 276.4 8.0085 296.4 0 1 0 18036 1,626,486.48

31-Dec-92 238.8 8.2052 276.4 0 0 1 18330 1,679,944.50

31-Mar-93 213.2 7.7332 238.8 0 0 0 17975 1,615,503.13

30-Jun-93 323.7 7.4515 213.2 1 0 0 18247 1,664,765.05

30-Sep-93 309.3 7.0778 323.7 0 1 0 18246 1,664,582.58

31-Dec-93 279.4 7.0537 309.3 0 0 1 18413 1,695,192.85

31-Mar-94 252.6 7.2958 279.4 0 0 0 18154 1,647,838.58

30-Jun-94 354.2 8.4370 252.6 1 0 0 18409 1,694,456.41

30-Sep-94 325.7 8.5882 354.2 0 1 0 18493 1,709,955.25

31-Dec-94 265.9 9.0977 325.7 0 0 1 18667 1,742,284.45

31-Mar-95 214.2 8.8123 265.9 0 0 0 18834 1,773,597.78

30-Jun-95 296.7 7.9470 214.2 1 0 0 18798 1,766,824.02

30-Sep-95 308.2 7.7012 296.7 0 1 0 18871 1,780,573.21

31-Dec-95 257.2 7.3508 308.2 0 0 1 18942 1,793,996.82

31-Mar-96 240 7.2430 257.2 0 0 0 19071 1,818,515.21

30-Jun-96 344.5 8.1050 240 1 0 0 19081 1,820,422.81

30-Sep-96 324 8.1590 344.5 0 1 0 19161 1,835,719.61

31-Dec-96 252.4 7.7102 324 0 0 1 19152 1,833,995.52

31-Mar-97 237.8 7.7905 252.4 0 0 0 19331 1,868,437.81

30-Jun-97 324.5 7.9255 237.8 1 0 0 19315 1,865,346.13

30-Sep-97 314.6 7.4692 324.5 0 1 0 19385 1,878,891.13

31-Dec-97 256.8 7.1980 314.6 0 0 1 19478 1,896,962.42

31-Mar-98 258.4 7.0547 256.8 0 0 0 19632 1,927,077.12

30-Jun-98 360.4 7.0938 258.4 1 0 0 19719 1,944,194.81

30-Sep-98 348 6.8657 360.4 0 1 0 19905 1,980,963.41

31-Dec-98 304.6 6.7633 348 0 0 1 20194 2,038,980.00

31-Mar-99 294.1 6.8805 304.6 0 0 0 20377 2,076,010.87

30-Jun-99 377.1 7.2037 294.1 1 0 0 20472 2,095,440.74

30-Sep-99 355.6 7.7990 377.1 0 1 0 20756 2,153,982.23

31-Dec-99 308.1 7.8338 355.6 0 0 1 21124 2,231,020.37

applied business forecasting and planning multiple regression analysis

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