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
Y  0  1 x1   2 x2  3 X 3   4 x4  
Formal Statement of the Model
 General regression model

Y  0  1 x1   2 x2     k xk  
• 0, 1, , k are parameters
• X1, X2, …,Xk are known constants
•  , the error terms are independent N(o, 2)
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

yˆ  b0  b1 x1  b2 x2    bk xk
to the data.
 The least-squares method chooses the b’s that
make the sum of squares of the residuals as small
as possible.
Estimating the parameters of the model
 The least-squares estimates are the values that
minimize the quantity
n

 i i
( y
i 1
 ˆ
y ) 2

 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.
Estimating the parameters of the model
 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
Estimating the parameters of the model
 The observed variability of the responses
about this fitted model is measured by the
variance n
1
S 
2

n  k  1 i 1
( yi  yˆ i ) 2

and the regression standard error

s  s2
Estimating the parameters of the model
 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:

 i
( y  y ) 2
  i
( ˆ
y  y ) 2
  i i
( y  ˆ
y ) 2

SST  SSR  SSE

 df:
n  1  k  (n  k  1)
Analysis of Variance Table

Source Sum of df Mean F-test

Squares Square

Regression SSR k MSR= MSR/MSE

SSR/k

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:
H 0 : 1   2     k  0
H a : not all  i (i  1, k ) equal zero
 We use the test statistic:
MSR
F
MSE
F-test for the overall fit of the model
 The decision rule at significance level  is:
 Reject H0 if
F  F ( ; k , n  k  1)
 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
Interval estimation of i
 For our regression model, we have:
bi   i
has a t - distributi on with n - k - 1 degrees of freedom
s (bi )
 Therefore, an interval estimate for i
with 1-  confidence coefficient is:

bi  t ( ; n  k  1) s (bi )
2
Where
MSE
s(bi ) 
 ( x  x )2
Significance tests for i
 To test:
H 0 : i  0
H a : i  0
 We may use the test statistic:
bi
t
s (bi )
 Reject H0 if

t  t ( ; n  k  1) or
2

t  t ( ; n  k  1)
2
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.
Multiple regression model Building
 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.
Multiple regression model Building
 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
SUBSCRIBE   0  1  ADRATE   2  APIPOP  3  COMPETE   4  SIGNAL
Example:Sales Forecasting
 Excel Summary output
SUMMARY OUTPUT

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

ANOVA
df SS MS F Significance F
Regression 4 1098857.84 274714.4601 13.44626923 7.52E-05
Residual 15 306458.0092 20430.53395
Total 19 1405315.85

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 51.42007002 98.97458277 0.51952803 0.610973806 -159.539 262.3795
AD_Rate -0.267196347 0.081055107 -3.296477624 0.004894126 -0.43996 -0.09443
Signal -0.020105139 0.045184758 -0.444954014 0.662706578 -0.11641 0.076204
APIPOP 0.440333955 0.135200486 3.256896248 0.005307766 0.152161 0.728507
Compete 16.230071 26.47854322 0.61295181 0.549089662 -40.2076 72.66778
Example:Sales Forecasting
 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.
Example:Sales Forecasting
 Excel Summary Output
SUMMARY OUTPUT

Regression Statistics
Multiple R 0.882638739
R Square 0.779051144
Adjusted R Square 0.737623233
Standard Error 139.3069743
Observations 20

ANOVA
df SS MS F Significance F
Regression 3 1094812.92 364937.64 18.80498277 1.69966E-05
Residual 16 310502.9296 19406.4331
Total 19 1405315.85

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 51.31610447 96.4618242 0.531983558 0.602046756 -153.1737817 255.806
AD_Rate -0.259538026 0.077195983 -3.36206646 0.003965102 -0.423186162 -0.09589
APIPOP 0.433505145 0.130916687 3.311305499 0.004412929 0.15597423 0.711036
Compete 13.92154404 25.30614013 0.550125146 0.589831583 -39.72506442 67.56815
Example:Sales Forecasting
 The variable Compete is the next variable to
get rid of.
 Example:Sales Forecasting
 Excel Summary Output
SUMMARY OUTPUT

Regression Statistics
Multiple R 0.8802681
R Square 0.774871928
Adjusted R Square 0.748386273
Standard Error 136.4197776
Observations 20

ANOVA
df SS MS F Significance F
Regression 2 1088939.802 544469.901 29.2562866 3.13078E-06
Residual 17 316376.0474 18610.35573
Total 19 1405315.85

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 96.28121395 50.16415506 1.919322948 0.07188916 -9.556049653 202.1184776
AD_Rate -0.254280696 0.075014548 -3.389751739 0.003484198 -0.41254778 -0.096013612
APIPOP 0.495481252 0.065306012 7.587069489 7.45293E-07 0.357697418 0.633265086
Example:Sales Forecasting

• All the variables in the model are

statistically significant, therefore our final
model is:
Final Model

SUBSCRIBE  96.28  0.25  ADRATE  0.495 APIPOP

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.
1
VIFj  , j  1,2,k
1 Rj
2

 R 2j Is the coefficient of determination from the

regression of the jth independent variable on the
remaining k-1 independent variables.
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 Forecasting
Pearson 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
 Example:Sales Forecasting

SUBSCRIBE  51.42  0.27  ADRATE - .02  SIGNAL  0.44  APIPOP 16.23 COMPETE

SUBSCRIBE  51.32  0.26  ADRATE  0.43 APIPOP 13.92  COMPETE

SUBSCRIBE  96.28  0.25  ADRATE  0.495 APIPOP

Example:Sales Forecasting
 VIF calculation:
 Fit the model
APIPOP  0  1  SIGNAL   2  ADRATE  3  COMPETE
SUMMARY OUTPUT

Regression Statistics
Multiple R 0.878054
R Square 0.770978
Adjusted R Square 0.728036
Standard Error 264.3027
Observations 20

ANOVA
df SS MS F Significance F
Regression 3 3762601 1254200 17.9541 2.25472E-05
Residual 16 1117695 69855.92
Total 19 4880295

Coefficients
Standard Error t Stat P-value Lower 95% Upper 95%
Intercept -472.685 139.7492 -3.38238 0.003799 -768.9402258 -176.43
Compete 159.8413 28.29157 5.649786 3.62E-05 99.86587622 219.8168
ADRATE 0.048173 0.149395 0.322455 0.751283 -0.268529713 0.364876
Signal 0.037937 0.083011 0.457012 0.653806 -0.138038952 0.213913
Example:Sales Forecasting
 Fit the model
Compete   0  1  ADRATE   2  APIPOP  3  SIGNAL
SUMMARY OUTPUT

Regression Statistics
Multiple R 0.882936
R Square 0.779575
Adjusted R Square 0.738246
Standard Error 1.34954
Observations 20

ANOVA
df SS MS F Significance F
Regression 3 103.0599 34.35329 18.86239 1.66815E-05
Residual 16 29.14013 1.821258
Total 19 132.2

Coefficients
Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 3.10416 0.520589 5.96278 1.99E-05 2.000559786 4.20776
ADRATE 0.000491 0.000755 0.649331 0.525337 -0.001110874 0.002092
Signal 0.000334 0.000418 0.799258 0.435846 -0.000552489 0.001221
APIPOP 0.004167 0.000738 5.649786 3.62E-05 0.002603667 0.005731
 Example:Sales Forecasting
 Fit the model
Signal   0  1  ADRATE   2  APIPOP  3  COMPETE
SUMMARY OUTPUT

Regression Statistics
Multiple R 0.512244
R Square 0.262394
Adjusted R Square 0.124092
Standard Error 790.8387
Observations 20

ANOVA
df SS MS F Significance F
Regression 3 3559789 1186596 1.897261 0.170774675
Residual 16 10006813 625425.8
Total 19 13566602

Coefficients
Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 5.171093 547.6089 0.009443 0.992582 -1155.707711 1166.05
APIPOP 0.339655 0.743207 0.457012 0.653806 -1.235874129 1.915184
Compete 114.8227 143.6617 0.799258 0.435846 -189.7263711 419.3718
ADRATE -0.38091 0.438238 -0.86919 0.397593 -1.309935875 0.548109
Example:Sales Forecasting
 Fit the model
ADRATE  0  1  Signal   2  APIPOP  3  COMPETE
SUMMARY OUTPUT

Regression Statistics
Multiple R 0.399084
R Square 0.159268
Adjusted R Square 0.001631
Standard Error 440.8588
Observations 20

ANOVA
df SS MS F Significance F
Regression 3 589101.7 196367.2 1.010346 0.413876018
Residual 16 3109703 194356.5
Total 19 3698805

Coefficients
Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 253.7304 298.6063 0.849716 0.408018 -379.2865355 886.7474
Signal -0.11837 0.136186 -0.86919 0.397593 -0.407073832 0.170329
APIPOP 0.134029 0.415653 0.322455 0.751283 -0.747116077 1.015175
Compete 52.3446 80.61309 0.649331 0.525337 -118.5474784 223.2367
Example:Sales Forecasting
 VIF calculation Results:
Variable R- Squared VIF
ADRATE 0.159268 1.19
COMPETE 0.779575 4.54
SIGNAL 0.262394 1.36
APIPOP 0.770978 4.36

 There is no significant multicollinearity.

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:
1 if stock company
x2 
0 otherwise
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:
y  0  1 x1   2 x2  
 Where:
 x1  size of firm
1 if stock company
 x2 
0 otherwise
Interpretation of Regression Coefficients
 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:
yˆi  b0  b1 x1  b2 (0)  b0 b1 x1 Mutual firms
 For a stock firm x2 = 1 and the response
function is:
yˆi  b0  b1 x1  b2 (1)  (b0  b2 )  b1 x1 Stock firms
Interpretation of Regression Coefficients
 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.
Interpretation of Regression Coefficients
 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 Mutual
26 92 0 Mutual
21 175 0 Mutual
30 31 0 Mutual
22 104 0 Mutual
0 277 0 Mutual
12 210 0 Mutual
19 120 0 Mutual
4 290 0 Mutual
16 238 1 Stock
28 164 1 Stock
15 272 1 Stock
11 295 1 Stock
38 68 1 Stock
31 85 1 Stock
21 224 1 Stock
20 166 1 Stock
13 305 1 Stock
30 124 1 Stock
14 246 1 Stock
Example: Insurance Innovation Adoption
 Fitting the regression model
y  0  1 x1   2 x2  
Where
 x1  size of firm
1 if stock company

x2 
0 otherwise

 fitted response function is:

yˆ  33.87  .1061x1  8.77 x2
Example: Insurance Innovation Adoption
SUMMARY OUTPUT

Regression Statistics
Multiple R 0.95993655
R Square 0.92147818
Adjusted R Square 0.91224031
Standard Error 2.78630562
Observations 20

ANOVA
df SS MS F Significance F
Regression 2 1548.820517 774.4103 99.75016 4.04966E-10
Residual 17 131.979483 7.763499
Total 19 1680.8

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 33.8698658 1.562588138 21.67549 8E-14 30.57308841 37.16664321
Size -0.10608882 0.007799653 -13.6017 1.45E-10 -0.122544675 -0.089632969
type of firm 8.76797549 1.286421264 6.815789 3.01E-06 6.053860079 11.4820909
Example: Insurance Innovation Adoption
 The fitted response function is:
yˆ  33.87  .1061x1  8.77 x2

 Stock firms response function is:

yˆ  (33.87  8.77)  .1061x1

 Mutual firms response function is:

yˆ  33.87  .1061x1
 Interpretation ?
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).
 0
50
100
150
200
250
300
350
400
Mar-90

Jul-90

Nov-90

Mar-91

1
Jul-91

Nov-91

Mar-92

1
Jul-92

Nov-92

1
Mar-93

Jul-93

Nov-93
1

Mar-94

Jul-94

Nov-94
1

Mar-95

Jul-95

Nov-95
Private Housing Starts (PHS) in Thousands of Units

Mar-96

Jul-96

Nov-96
"1" marks the first quarter of each year.
1

Mar-97

Jul-97

Nov-97

Mar-98
1

Jul-98

Nov-98
Example: Private Housing Starts (PHS)
Example: Private Housing Starts (PHS)
 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.
Example: Private Housing Starts (PHS)
 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 0
30-Jun-90 271.3 10.3372 1 0 0
30-Sep-90 233 10.1033 0 1 0
31-Dec-90 173.6 9.9547 0 0 1
31-Mar-91 146.7 9.5008 0 0 0
30-Jun-91 254.1 9.5265 1 0 0
30-Sep-91 239.8 9.2755 0 1 0
31-Dec-91 199.8 8.6882 0 0 1
31-Mar-92 218.5 8.7098 0 0 0
30-Jun-92 296.4 8.6782 1 0 0
30-Sep-92 276.4 8.0085 0 1 0
31-Dec-92 238.8 8.2052 0 0 1
31-Mar-93 213.2 7.7332 0 0 0
30-Jun-93 323.7 7.4515 1 0 0
30-Sep-93 309.3 7.0778 0 1 0
31-Dec-93 279.4 7.0537 0 0 1
31-Mar-94 252.6 7.2958 0 0 0
30-Jun-94 354.2 8.4370 1 0 0
30-Sep-94 325.7 8.5882 0 1 0
31-Dec-94 265.9 9.0977 0 0 1
31-Mar-95 214.2 8.8123 0 0 0
30-Jun-95 296.7 7.9470 1 0 0
30-Sep-95 308.2 7.7012 0 1 0
31-Dec-95 257.2 7.3508 0 0 1
31-Mar-96 240 7.2430 0 0 0
30-Jun-96 344.5 8.1050 1 0 0
30-Sep-96 324 8.1590 0 1 0
31-Dec-96 252.4 7.7102 0 0 1
31-Mar-97 237.8 7.7905 0 0 0
30-Jun-97 324.5 7.9255 1 0 0
30-Sep-97 314.6 7.4692 0 1 0
31-Dec-97 256.8 7.1980 0 0 1
31-Mar-98 258.4 7.0547 0 0 0
30-Jun-98 360.4 7.0938 1 0 0
30-Sep-98 348 6.8657 0 1 0
31-Dec-98 304.6 6.7633 0 0 1
31-Mar-99 294.1 6.8805 0 0 0
30-Jun-99 377.1 7.2037 1 0 0
30-Sep-99 355.6 7.7990 0 1 0
31-Dec-99 308.1 7.8338 0 0 1
Example: Private Housing Starts (PHS)
 The regression model for private housing
starts (PHS) is:
PHS   0  1 (MR)   2 (Q2)  3 (Q3)   4 (Q4)
 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.
Example: Private Housing Starts (PHS)
SUMMARY OUTPUT

Regression Statistics
Multiple R 0.885398221
R Square 0.78393001
Adjusted R Square 0.759236296
Standard Error 26.4498851
Observations 40

ANOVA
df SS MS F Significance F
Regression 4 88837.93624 22209.48406 31.74613731 3.33637E-11
Residual 35 24485.87476 699.5964217
Total 39 113323.811

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 473.0650749 35.54169837 13.31014264 2.93931E-15 400.9115031 545.2186467
MR -30.04838192 4.257226391 -7.058206249 3.21421E-08 -38.69102153 -21.40574231
Q2 95.74106935 11.84748487 8.081130334 1.6292E-09 71.689367 119.7927717
Q3 73.92904763 11.82881519 6.249911462 3.62313E-07 49.91524679 97.94284847
Q4 20.54778131 11.84139803 1.73524961 0.091495355 -3.491564078 44.5871267
Example: Private Housing Starts (PHS)
 Use the prediction equation to make a
forecast for each of the fourth quarter of
1999.
 Prediction equation:
PHSˆ  473.06  30.05( MR )  95.74(Q 2)  73.93(Q3)  20.55(Q 4)
Example: Private Housing Starts (PHS)
400

Private Housing Starts (PHS) with a Simple Regression Forecast (PHSF1) and a Multiple Regression Forecast (PHSF2) in
Thousands of Units
350

300

250

200

150

100

50

0
Jul-90

Jul-91

Jul-92

Jul-93

Jul-94

Jul-95

Jul-96

Jul-97

Jul-98
Mar-90

Mar-91

Mar-92

Mar-93

Mar-94

Mar-95

Mar-96

Mar-97

Mar-98
Nov-90

Nov-91

Nov-92

Nov-93

Nov-94

Nov-95

Nov-96

Nov-97

Nov-98
PHS PHSF1 PHSF2
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:
yt   0  1 xt   t
 Where
 t    t 1  t
 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
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
yt   0  1 xt   t
 t    t 1  t
 The hypothesis to be tested are:
H0 :   0
Ha :   0
 The alternative hypothesis is  > 0 since in
business and economic time series tend to show
positive correlation.
Durbin-Watson Test for Serial
Correlation
 The Durbin-Watson statistic is defined as
n

 (e t  et 1 ) 2
DW  t 2
n

e
t 1
2
t

 Where
et  yt  yˆt  the residual for time period t
et 1  yt 1  yˆt 1  the residual for time period t -1
Durbin-Watson Test for Serial
Correlation
 The auto correlation coefficient  can be
estimated by the lag 1 residual
autocorrelation r1(e)
n

e e t t 1
r1 (e)  t 2
n

 t
e 2

t 1

 And it can be shown that

DW  2(1  r1 (e))
Durbin-Watson Test for Serial
Correlation
 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.
Example
Year Quarter t CompSale InduSale
1983 1 1 20.96 127.3
2 2 21.4 130
3 3 21.96 132.7
4 4 21.52 129.4
1984 1 5 22.39 135
2 6 22.76 137.1
3 7 23.48 141.2
4 8 23.66 142.8
1985 1 9 24.1 145.5
2 10 24.01 145.3
3 11 24.54 148.3
4 12 24.3 146.4
1986 1 13 25 150.2
2 14 25.64 153.1
3 15 26.36 157.3
4 16 26.98 160.7
1987 1 17 27.52 164.2
2 18 27.78 165.6
3 19 28.24 168.7
4 20 28.78 171.7
Example
Blaisdell Company Example

35
30
Company Sales ($

25
millions)

20
15
10
5
0
0 50 100 150 200
Industry sales($ millions)
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:
H0 :   0
Ha :   0

 The test statistic is:

n

 (e t  et 1 ) 2
DW  t 2
n

e
t 1
2
t
Example
Year Quarter t Company sales(y) Industry sales(x) et et -et-1 (et -et-1)^2 et ^2
1983 1 1 20.96 127.3 -0.02605 0.000679
2 2 21.4 130 -0.06202 -0.03596 0.001293 0.003846
3 3 21.96 132.7 0.022021 0.084036 0.007062 0.000485
4 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 0.002169
2 6 22.76 137.1 0.046377 -0.00019 3.76E-08 0.002151
3 7 23.48 141.2 0.043617 -0.00276 7.61E-06 0.001902
4 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 0.008911
2 10 24.01 145.3 -0.14914 -0.05474 0.002997 0.022243
3 11 24.54 148.3 -0.14799 0.001152 1.33E-06 0.021901
4 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 0.000526
2 14 25.64 153.1 0.105852 0.12878 0.016584 0.011205
3 15 26.36 157.3 0.085464 -0.02039 0.000416 0.007304
4 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 0.000848
2 18 27.78 165.6 0.042316 0.013204 0.000174 0.001791
3 19 28.24 168.7 -0.04416 -0.08648 0.007478 0.00195
4 20 28.78 171.7 -0.03301 0.011152 0.000124 0.00109

0.097941 0.133302
Blaisdell Company Example

35
30
Example
.09794
DW   .735
.13330
 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.
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.
Extensions of the Multiple Regression
Model
 Some common forms of nonlinear functions
are :
Y   0  1 ( X )   2 ( X 2 )
Y   0  1 ( X )   2 ( X 2 )   3 ( X 3 )

Y   0  1 (1 X )

Y  e  0 X 1
Extensions of the Multiple Regression
Model
 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
PHS   0  1 (MR)   2 (Q2)  3 (Q3)   4 (Q4)
 Where MR is the mortgage rate and Q2, Q3, and Q4 are
indicators variables for quarters 2, 3, and 4.
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:
PHS   0  1 (MR)   2 (Q2)  3 (Q3)   4 (Q4)  5 ( DPI )

 Regression results for this model are shown

in the next slide.
Example: Private Housing Start
SUMMARY OUTPUT

Regression Statistics
Multiple R 0.943791346
R Square 0.890742104
Adjusted R Square 0.874187878
Standard Error 19.05542121
Observations 39

ANOVA
df SS MS F Significance F
Regression 5 97690.01942 19538 53.80753 6.51194E-15
Residual 33 11982.59955 363.1091
Total 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 182.9546249
MR -20.1992545 4.124906847 -4.8969 2.5E-05 -28.59144723 -11.80706176
Q2 97.03478074 8.900711541 10.90191 1.78E-12 78.9261326 115.1434289
Q3 75.40017073 8.827185877 8.541813 7.17E-10 57.44111179 93.35922967
Q4 20.35306822 8.83373887 2.304015 0.027657 2.380677107 38.32545934
DPI 0.022407799 0.004356973 5.142974 1.21E-05 0.013543464 0.031272134
Example: Private Housing Start
 The prediction model is
PHSˆ  31.06  20.19( MR )  97.03(Q 2)  75.40(Q3)  20.35(Q 4)  0.02( DPI )

 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.
Example: Private Housing Start
 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.
Example: Private Housing Start
 The Plot of PHS against
DPI shows a curve linear Private Housing Start and Disposable Personal Income

relation. 21500

21000

 Next we introduce a 20500

nonlinear term into the 20000

regression.

PHS
19500

19000

 The square of disposable 18500

personal income per capita 18000

(DPI2) is included in the 17500

0 50 100 150 200

DPI
250 300 350 400

regression model.
 Example: Private Housing Start
 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:
PHS   0  1 ( MR )   2 (Q 2)   3 (Q3)   4 (Q4)   5 ( DPI )   6 ( DPI 2 )   7 ( LPHS )

 Regression results for this model are shown

in the next slide.
Example: Private Housing Start
SUMMARY OUTPUT

Regression Statistics
Multiple R 0.97778626
R Square 0.956065971
Adjusted R Square 0.946145384
Standard Error 12.46719572
Observations 39

ANOVA
df SS MS F Significance F
Regression 7 104854.2589 14979.17985 96.37191 3.07085E-19
Residual 31 4818.360042 155.4309691
Total 38 109672.619

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 716.5926532 1017.664989 0.704153784 0.486593 -1358.949934 2792.13524
MR -13.65521724 3.093504134 -4.414158396 0.000114 -19.96446404 -7.345970448
Q2 106.9813297 6.069780998 17.62523718 1.04E-17 94.60192287 119.3607366
Q3 27.72122303 9.111432565 3.042465916 0.004748 9.138323433 46.30412262
Q4 -13.37855186 7.653050858 -1.748133144 0.09034 -28.98706069 2.22995698
DPI -0.060399279 0.104412354 -0.578468704 0.567127 -0.273349798 0.15255124
DPI SQUARED 0.000335974 0.000536397 0.626354647 0.535668 -0.000758014 0.001429963
LPHS 0.655786939 0.097265424 6.742241114 1.51E-07 0.457412689 0.854161189
Example: Private Housing Start
 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 Start
PERIOD 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