Business Quantitative Lecture 3

Quantitative Analysis for BusinessLecture 3July 12th, 2010Saksarun (Jay) Mativachranon

Assumptions of the Regression ModelIf we make certain assumptions about the errors in a regression model, we can perform statistical tests to determine if the model is useful Errors are independentErrors are normally distributedErrors have a mean of zeroErrors have a constant varianceA plot of the residuals (errors) will often highlight any glaring violations of the assumption

Residual Plots ErrorXA random plot of residualsFigure 4.4A

Residual Plots ErrorXNonconstant error varianceFigure 4.4B

Residual Plots ErrorXNonlinear relationshipFigure 4.4C

Analysis of variance (Anova)Analysis of Variance (ANOVA)A statistical procedure for analyzing the total variability of a data set

Analysis of variance (Anova)Sum of squares total (SST)Measures the total variation in the dependent variableSum of squares of regression (SSR)Measures the variation in the dependent variable explained by the independent variableSum of squares of errors (SSE)Measures the unexplained variation

Estimating the VarianceErrors are assumed to have a constant variance ( 2), but we usually don’t know thisIt can be estimated using the mean squared error (MSE), s2wheren = number of observations in the samplek = number of independent variables

The Standard error of estimateThe standard error of estimate (SEE) or The standard error of the regressionMeasures uncertainty between independent and dependent variables

The f statisticF-test asses how well a set of independent variables, as a group, explains the variation in the dependent variableWhereMSR = mean regression sum of squaresMSE = mean squared errork = the number of slope parameters (k = 1 for linear regression)n = number of observations

F-statistic linear regressionFor linear regression, the hypotheses for the validity of the model are;H0: b1 = 0Ha: b1 ≠ 0To determine if b1 is statistically significant, the calculated F-statistic is compared with the critical F-value, Fc, at the appropriate level of significance.

F-statistic linear regressionThe degree of freedom (df) for the numerator and denominator with one independent variable are;dfnumerator = k = 1dfdenominator = n – k – 1 = n – 2Decision for F-testReject H0 if F > Fc

Estimating the VarianceFor Company AWe can estimate the standard deviation, s

This is also called the standard error of the estimate or the standard deviation of the regressionTesting the Model for SignificanceWhen the sample size is too small, you can get good values for MSE and r2 even if there is no relationship between the variablesTesting the model for significance helps determine if the values are meaningfulWe do this by performing a statistical hypothesis test

Testing the Model for SignificanceWe start with the general linear modelIf 1 = 0, the null hypothesis is that there is no relationship between X and Y

The alternate hypothesis is that there is a linear relationship (1≠ 0)

If the null hypothesis can be rejected, we have proven there is a relationship

We use the F statistic for this testTesting the Model for SignificanceThe F statistic is based on the MSE and MSRwherek = number of independent variables in the modelThe F statistic is

This describes an F distribution with degrees of freedom for the numerator = df1 = k degrees of freedom for the denominator = df2 = n – k – 1

Testing the Model for SignificanceIf there is very little error, the MSE would be small and the F-statistic would be large indicating the model is usefulIf the F-statistic is large, the significance level (p-value) will be low, indicating it is unlikely this would have occurred by chanceSo when the F-value is large, we can reject the null hypothesis and accept that there is a linear relationship between X and Y and the values of the MSE and r2 are meaningful

Steps in a Hypothesis TestSpecify null and alternative hypothesesSelect the level of significance (). Common values are 0.01 and 0.05Calculate the value of the test statistic using the formula

Steps in a Hypothesis TestReject the null hypothesis if the test statistic is greater than the F-value from the table Otherwise, do not reject the null hypothesis:Reject the null hypothesis if the observed significance level, or p-value, is less than the level of significance (). Otherwise, do not reject the null hypothesis:Make a decision using one of the following methods

Step 3. Calculate the value of the test statisticCompany AStep 1.H0: 1 = 0 (no linear relationship between X and Y)H1: 1≠ 0 (linear relationship exists between X and Y)Step 2. Select  = 0.05

Step 4. Reject the null hypothesis if the test statistic is greater than the F-valuedf1 = k = 1df2 = n – k – 1 = 6 – 1 – 1 = 4 The value of F associated with a 5% level of significance and with degrees of freedom 1 and 4 isF0.05,1,4 = 7.71Fcalculated = 9.09Reject H0 because 9.09 > 7.71Company A

0.05F = 7.719.09Company AWe can conclude there is a statistically significant relationship between X and Y

The r2 value of 0.69 means about 69% of the variability in sales (Y) is explained by Man Hour (X)Limitation of regression analysisLinear relationships can change over timeThis is referred to as parameter instabilityEven if the model is accurate, its usefulness will be limited if other market participants are also aware of and act on this modelIf the assumptions do not hold, the interpretation and tests of hypotheses may not be valid

Business Quantitative Lecture 3

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Business Quantitative Lecture 3