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Quantitative Methods
Review—Bivariate Regression
What is the criterion that OLS uses to “fit” a line to your data?
What is a parameter? A parameter estimate?
What are independent variables—or rhs (right-hand-side)
variables? Dependent variables?
What is the slope? The intercept? The error term (in the
population) or the residual (in the sample)?
Review—Bivariate Regression
Review of notation—slope, intercept, error (estimated
or sample VS. true or population)
Two possible consequences of violating an OLS
assumption—”bias” (what does that mean?) and
inflated / deflated standard errors (what does that
mean?)
Review—Bivariate Regression
Assumptions:
No measurement error
Specification—include all relevant rhs variables, no
irrelevant rhs variables, linear relationship
Is this likely what our data look like?
Homeskedastic error terms (no heteroskedasticity)
No autocorrelation
Bivariate Regression: Robustness
A discussion about standard errors, p-values, ad
statistical significance
Confidence intervals. What are they?
Confidence intervals are a range in which you
would expect the true parameter to fall a
pre-specified percentage of the time.
Bivariate Regression: Robustness
The wider the confidence interval, the less certain you are of
the estimate. (A relatively wide confidence interval means
that you could gather another sample, and would not be
confident that your new slope estimate would be relatively
close to the one you have from this sample).
The wider the confidence interval....
The higher the p-value (farther away from .05 or .01)...
The less statistically significant the results....
The less confident you are that there is a non-zero effect of the
independent variable on the dependent variable
Bivariate Regression: Robustness
The narrower the confidence interval, the more
“robust”, “efficient”, “stable” the results are.
If you were to gather an infinite number of
samples from your population, and calculate
an infinite number of slope estimates (one
from each sample), you dno’t expect that
the slope estimate will change much from
sample to sample...
Bivariate Regression: Robustness
The standard error and variance of the slope
estimate....
is a closely related concept. The larger the
standard error, the less confident you are in
your results (and the wider your confidence
interval). Recall the image of the seesaw.
Bivariate Regression: Robustness
Let’s start with the variance of the residual. The formula
for the variance of the residual (or the estimated
variance of the error term):

 
 e e
2
i

ˆ  

 n2

2


Bivariate Regression: Robustness
Note two elements of that equation-First, what (by assumption) is the average
residual?
And second, why are we subtracting 2 from
our sample size?
Bivariate Regression: Robustness
What is the variance of the slope estimate?
 ˆ
2
2



 Xi  x




2



Bivariate Regression: Robustness
Note the numerator of the variance of the slope
estimate b: it taps into the variance of the residuals
(or, how well the data “fit” your estimated line)
Note the denominator of the variance of the slope
estimate b: it taps into the range or variance of X.
Bivariate Regression: Robustness
So, if the data fit your line well....
The numerator of that equation is reduced
The variance of the slope estimate b is reduced; your results are
more stable
The confidence interval for β is narrower
Your results are more statistically significant; your p-value is
relatively low.
You are more likely to reject the null hypothesis that β=0
Bivariate Regression: Robustness
And, if you have relatively good variance in X...
The denominator of that equation is increased
The variance of the slope estimate b is reduced; your results are
more stable
The confidence interval for β is narrower
Your results are more statistically significant; your p-value is
relatively low.
You are more likely to reject the null hypothesis that β=0
Bivariate Regression: Robustness
The equation above is for the variance of your estimated
slope b.
Your computer printouts will generally give you the
standard error of b.
How do we calculate a standard error based on a
variance?
Bivariate Regression: Robustness
The confidence interval for b is analogous to the
variance and standard error, as noted above...
 
 ˆ  n  2  ˆ


t
*





  2  ˆ
 
,  ˆ   t n  2  * ˆ
     ˆ
  2 



Bivariate Regression: Robustness
So, the slope estimate plus / minus
The t-value * the standard error of the slope estimate
What is α? (It is 100-CL. Our CL is predetermined).
Why are we dividing α by 2?
Where can we find the t-value?
How do we interpret a confidence interval?
Bivariate Regression: T values
Recall the central limit theorem, which said that for any
population with known mean μ and known variance σ2,
random samples can be drawn, and the means of these
samples will be
2

x  N ( ,
)
n
Bivariate Regression: T values
We use the t distribution in probability testing. Suppose X is
some Random Variable with a true mean of μ and a true
variance of σ2. Of course, in “real life”, we never know
these ‘true” values. We estimate μ with
x
And we don’t know σ2, so we estimate it with s2. So instead of
saying that we approximate a normal distribution, we say....
Bivariate Regression: T values
And we don’t know σ2, so we estimate it with s2. So instead of
saying that we approximate a normal distribution, we say....
x  
s/
n
~ Tn-1
Bivariate Regression: T values
As n gets larger, T distribution is closer to N(0,1) distribution; the
mean of the t distribution is always 0, and as n increases,
the variance of the t-distribution shrinks to 1.
x  
s/
n
~ Tn-1
Bivariate Regression: T values
Our t value is calculated by setting μ to a hypothesized value
(usually 0), and then taking our sample estimate, and
dividing it by the standard error. Note that this corresponds
to the formula below (although note that instead of the
mean of X, we will be using “b” as our sample / estimated
slope)
x  
s/
n
~ Tn-1
Bivariate Regression: Hypothesis
Testing
If our 1- α confidence interval includes zero, then we do
not reject (we “fail to reject” our null hypothesis
H0: β=0 at the 1- α level (2 tailed test).
Of course, if our 1- α CI does not include zero, then we
accept H1: β ≠ 0 (we do reject H0: β=0) at the 1- α
Confidence Level.
Bivariate Regression: Prob-Values
Prob values for slope coefficient are analogous to
confidence intervals. Most computer packages will
report these p-values for each slope coefficient. The
universal decision rule indicates that we
Reject H0: β=0 with (1-α) confidence if p-value < α
Bivariate Regression: Prob-Values
Pre-Determined
CI
1-α
α
reported
p-value
Is p-value
< α?
Conclusion
90%
1-.10
.10
.0374
Yes
Reject Ho
95%
1-.05
.05
.0374
Yes
Reject Ho
98%
1-.02
.02
.0374
No
Fail to Reject Ho
99%
1-.01
.01
.0374
No
Fail to Reject Ho
Bivariate Regression: One-tailed
versus two-tailed tests.
We use one-tailed tests when we have a directional
hypothesis.
One tailed tests make parameter estimates more
significant, because you are restricting H1 to a
narrower set of possibilities.
In confidence intervals, the α remains the same,
because you’ve picked a pre-determined
confidence level—but you can think of the p-value
(the area under the curve that represents greater
than t) as being halved.
Bivariate Regression: Summary
We are estimating slopes and intercepts—and so we talk
about the degree of confidence we have, based on
our sample slopes and intercepts.
That concept of “confidence”, “robustness”,
“efficiency”, “stability” is part of inferential statistics.
And, in general, a better fit of the data to the model—
and more variance in the explanatory / independent
variables – tends to make the findings more robust.
Bivariate Regression: Summary
This makes sense—if you only ask a couple of people
who they are voting for in the Democratic /
Republican primaries, you will not have much
variance—and you would not be confident if you
tried to generalize to a larger population.
And research problems where there isn’t much variance
in the independent variables (or dependent variable),
and where the dependent variable is a “rare event”
are just inherently difficult to predict (although there
are ways to weight the observations so that one can
address those issues).
Bivariate Regression: Summary
Likewise, we are always going to be thinking about two
possible problems—we can have deflated or inflated
standard errors if we are violating OLS assumptions (so,
our results are more or less significant than they would
otherwise be).
Or our results are biased, which means that the
estimated slope would not average out to the true
slope, even if one collected an infinite number of
samples, and an infinite number of estimate slopes.
Bivariate Regression: Summary
These concepts—of confidence and bias—also carry
over to all inferential methods.
And, keep in mind that there is a difference between
statistical significance (as signaled by p-values or tvalues or confidence intervals) and the magnitude of
b.
You may have a very small effect, but it is “statistically
significant” because it is very robust (remember what
goes into the t--
Bivariate Regression: Summary
You may have a very small effect, but it is “statistically
significant” because it is very robust (remember what goes
into the t—the value of b, divded by the standard error of
b).
Or, you may have a very large effect, but cannot conclude
that it is different from 0, because it is not very robust—
you’re not that sure it would be large if you collected a
different sample.
These concepts, too, carry over across methods. It is very
important to interpret both statistical significance and
magnitude, and to recognize they are not the same.
Bivariate Regression: Residuals
True Model:
yi = σ + βxi + εi
Which is estimated with:
Yi = a + bXi + ei
εi is the true error term for observation i. “e” is the estimated
error (residual) for observation i.
Bivariate Regression: Residuals
So,
ei = yi – (a + bxi)
Or
ei = observed Y – predicted Y
Think of the error term in the population as not an error, but as a
disturbance or a stochastic shock, whose deviation from the “true”
population line is due to randomness.
Bivariate Regression: R2
Notice that observed Y – Mean Y = total deviation of Yi
from mean Y.
Notice that predicted Y – mean of Y is the deviation of Yi
from the mean of Y explained by OLS regression line
And notice that observed Y – predicted Y is the
remaining unexplained deviation of Yi from mean Y
(error)
Bivariate Regression: R2
Case
(i)
Ordering
1
P<O<M
2
O<P<M
3
P<M<O
4
O<M<P
5
M<P<O
6
M<O<P
Total
(observed-mean)
Explained
(predicted - mean)
Unexplained
(Error)
(observed-predicted)
Bivariate Regression: R2
Of course, in any sample we have n data points—
so we’ll have n total deviations
And n explained deviations
And n unexplained deviations
Bivariate Regression: R2
Suppose we square each individual total, explained, and
unexplained deviation.
And then we sum up all of the squared total deviations, do the
same for all the squared explained deviations, and the
same for all the squared unexplained deviations.
We would see that
Sum of the Squared Total Deviations =
Sum of the Squared Explained Deviations +
Sum of the Squared Unexplained Deviations
Bivariate Regression: R2
OR,
TSS = Total Sum of Squares =
RSS (Regression Sum of Squares / Explained Sum of
Squares)
+
ESS (Error Sum of Squares / Residual Sum of Squares)
Bivariate Regression: R2
And R2 = RSS / TSS
(Note that this is the same as 1 – proportion of total
deviation (TSS) of Y from the mean that is
“unexplained” by OLS)
So, if R2 = .34, we can say that 34% of the total variation
in Y has been accounted for by the OLS regression of
Y and X
Bivariate Regression: R2
Why is R2 useful?
What are the limits of R2?
It is not really a measure of magnitude of the effect.
It is a measure of correlation, and so it depends in part
on the standard deviation of X and Y—and cannot be
compared across samples.
Models with high R2 are not necessarily “good”—and
models with low R2 are not necessarily “bad”.
Bivariate Regression: R2
It is not really a measure of magnitude of the effect.
It is a measure of correlation, and so it depends in part on the
standard deviation of X and Y—and cannot be compared
across samples.
Models with high R2 are not necessarily “good”—and models
with low R2 are not necessarily “bad”.
R2 can be biased, particularly in small samples.
And it can be a reflection of the number of variables on the
left hand sides (although there are ways to account for
this—adjusted R2)
Bivariate Regression: R2
The bottom line? – R2 is a measure of goodness of
fit, and as such can be useful. It is not a measure
of how good your results are.
And, when you think about it, what the R2 is doing is
telling you how well the data fit the line—how
good your prediction is compared to just using
the mean. The mean isn’t a great predictor of Y,
so the utility of the R2 is limited.
Bivariate Regression: Standard
Error of Estimate
ˆ 


2
ei
n2
Bivariate Regression: Some useful
equations....
( X  X )(Y  Y )

b
( X  X )
i
i
2
i
a  Y  bX