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Univariate Regression Variance, Slope & Correlation. MSIT3000 Lecture 18. Objectives. Learn how to estimate the disturbance in a regression model. Assess the usefulness of an OLS model through the slope: using hypothesis tests, and confidence intervals

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objectives
Objectives
  • Learn how to estimate the disturbance in a regression model.
  • Assess the usefulness of an OLS model through the slope:
    • using hypothesis tests, and
    • confidence intervals
  • Compare correlation & covariance to OLS.

Text: Chapter 9, sections 4 through 7; 2.9 & 2.10.

estimating var 2
Estimating Var() = 2
  • We use the observed error term (e) to estimate the disturbance [or predicted error term] ().
      • s2 = SSE/(n-2)
      • Remember, SSE = (y-yhat)2
      • We divide by n-2 because that is how many degrees of freedom left after we estimated the intercept and the slope.
interpreting s 2
Interpreting s2
  • Use the Empirical rule. We would expect “most” observations of y to be within two standard deviations (2s) of our prediction (yhat).
  • s =  s2 has several names. The text calls it “the estimated standard error of the regression model”. SAS calls it Root MSE (for Mean Squared Error); see the SAS output on p 498. s2 is called “the estimated variance”.
when does the slope tell us anything at all
When does the slope tell us anything at all ?
  • The model is:
    • Y = 0 + 1*X + 
  • If X has no impact on Y, the slope must be zero.
how can we test whether or not the model is useful
How can we test whether or not the model is useful?
  • We perform a hypothesis test to find out if the data suggest the slope is NOT equal to zero.
  • This is the default output from most statistical software.
  • What do we need to know in order to perform this Hypothesis Test?
    • The distribution of the slope-estimate under the null hypothesis.
the distribution of 1 hat
The distribution of 1-hat
  • If we know the variance of the disturbance, the variance of the slope is:
    • Var(1-hat) = ²/SS(xx)
  • In that case, the distribution of 1–hat would be:
    • 1-hat ~ N(1, /[SS(xx)] )
the distribution of 1 hat realistically
The distribution of 1-hat (realistically)
  • When we don’t know the standard error, we have to estimate the standard deviation of the disturbance using s:
    • s² = SSE/(n-2)  s = (SSE/[n-2])
  • And our test statistic is t distributed
    • just like our small sample tests for means.
testing whether x impacts y
Testing whether X impacts Y:
  • We want the burden of proof on the model, so the hypotheses are:
    • H0: 1 = 0
    • H1: 1  0
  • The second step is to find the rejection region (RR):
    • Our test statistic is t-distributed with n-2 degrees of freedom, therefore:
    • RR = < -  , - t/2 ]  [t/2 ,  >
testing whether x impacts y continued
Testing whether X impacts Y continued:
  • Step III: Calculate the test statistic.
    • TS = (1-hat – 0)/S 1-hat
      • Remember, under the null hypothesis, 1 = 0. If you wished to test the null hypothesis that
        • 1 = 1 you would subtract 1 in the numerator above.
      • You divide by the standard deviation of 1-hat:
        • S 1-hat = S/[SS(xx)]
        • where S2 = SSE/[n-2]
  • Step IV: Conclude.
p values and confidence intervals
P-values and Confidence Intervals
  • When you perform a hypothesis test you can also calculate the p-value, just as you would for a small-sample HT for a mean.
  • And you can also create a Confidence Interval (CI):
    • CI = 1-hat ± t/2 * S1-hat
objective 3 alternative measures of linear relationship
Objective 3: Alternative measures of linear relationship
  • We will now consider:
    • Covariance
    • Correlation
    • The Coefficient of Determination
covariance
Covariance
  • Covariance measures how much to variables “move around together”:
covariance matrix
Covariance Matrix

This is extremely useful in presenting how stocks relate to one another!

the correlation coefficient
The Correlation Coefficient
  • The Pearson moment coefficient of correlation
    • (or simply the “correlation coefficient”):
      • r = SS(xy)/[SS(xx)*SS(yy)]
      • -1  r  1
  • Note that b1 has the same numerator [i.e. SS(xy)], so if the slope is zero, the correlation coefficient is also zero.
  • r is an estimator for the linear correlation between x and y in the population:
      •  [rho]
correlation coefficient and covariance
Correlation Coefficient and Covariance
  • These two measure the same thing, but correlation is bound between –1 and 1:
the coefficient of determination
The Coefficient of Determination
  • The Coefficient of Determination is a measure of how much of the variation in y is explained by x.
  • The Coefficient of Determination will be useful also when we have multiple x’s to explain y. This is not true of the correlation coefficient.
if x explained nothing
If x explained nothing...
  • ...what would be the relationship between SS(yy) and SSE?
      • SS(yy) =  ( y – ybar )²
      • SSE =  ( y – yhat )²
if x explained nothing19
If x explained nothing...
  • ...what would be the relationship between SS(yy) and SSE?
      • SS(yy) =  ( y – ybar )²
      • SSE =  ( y – yhat )²
  • If x explains nothing, the best predictor for y, regardless of the value of x, would be ybar.
  • Therefore, if x explains nothing, we would expect:
      • SS(yy) = SSE
      • and if x explains very little, we would expect SS(yy)SSE.
a measure of how much x explains
A measure of how much x explains:
  • Out of the total variation in y [SS(yy)], a measure of how much x explains is:
      • SS(yy) – SSE [this is explained variation]
  • But because this does not have a useful scale, we calculate the proportion explained:
      • r² = [SS(yy) – SSE]/SS(yy)
      • r² = 1 – [SSE/SS(yy)]
  • Note: 0 r²  1
conclusion
Conclusion
  • Objectives addressed:
      • Learn how to estimate the error in a regression model.
      • Assess the usefulness of an OLS model through the slope:
        • using hypothesis tests, and
        • confidence intervals
    • Compare correlation & covariance to OLS.
  • Problems:
    • Text: 9.24; 9.31a, 9.34, 9.39, (9.54)
    • Exam 3A, 7-9 & 17-21.