Question

how do we calculate the standard errors of regression coefficients in multiple linear regression, that is, the standard error of the constant(B0), the standard error of B1 and standard error of B2

Answer #1

Let be a vector of order n X 1
containing the values of x1, be a vector of order n X 1 containing
the values of x2, be a vector of order n X 1 containing
the values of y and, lastly, be a vector containing "n" number of
1's and, as a result, of order n X 1 (required for the intercept
term).

Now, the design matrix is given as: , which is order n X 3.

The 3 parameters (2 regression coefficients and 1 intercept),
represented by , are estimated in the following way:
.

The SSE is now calculated as: .

Now, MSE = SSE/n - 3 = . This is
our residual standard error.

Now, the standard errors of the regression coefficients are given
in the following way:

.

This will resulting in a diagonal matrix, where the square root of
the diagonals are the standard errors of the constant, B1 and B2
respectively.

Using the OLS estimator, how do we derive the formula for the
slope coefficients (B1* , B2*) of the
regression line Yi= Bo* +
B1*Xi + B2*Zi ?

Suppose that your linear regression model includes a constant
term, so that in the linear regression model
Y = Xβ + ε
The matrix of explanatory variables X can be
partitioned as follows: X = [i X1]. The
OLS estimator of β can thus be partitioned
accordingly into b’ = [b0
b1’], where b0 is
the OLS estimator of the constant term and
b1 is the OLS estimator of the slope
coefficients.
a) Use partitioned regression to derive formulas for...

how do we get error sum of squares in a multiple
linear regression with two independent variables x1 and x2

how to calculate standard error of multiple regression
coefficient by hand??
In simple regression, we have a clear formula of it, but I am
not sure there is formula for multiple regression.
Like how to get the standard error for bata-5 something?? Is
that even possible to get it by hand like simple regression??
Please explain it in detail. Thanks

True or False. Explain your answer:
d) Least squares estimates of the regression coefficients b0,
b1, . . . bn are chosen to maximize R2 .
e) If all the explanatory variables are uncorrelated, the
variance inflation factor (VIF) for each explanatory variable will
be 1.
) b0 and b1 from a simple linear regression model are
independent.

In the simple linear regression model estimate Y =
b0 + b1X
A. Y - estimated average predicted value, X –
predictor, Y-intercept (b1), slope
(b0)
B. Y - estimated average predicted value, X –
predictor, Y-intercept (b0), slope
(b1)
C. X - estimated average predicted value, Y –
predictor, Y-intercept (b1), slope
(b0)
D. X - estimated average predicted value, Y –
predictor, Y-intercept (b0), slope
(b1)
The slope (b1)
represents
A. the estimated average change in Y per...

Multiple Linear Regression
We consider the misspecification problem in multiple linear
regression. Suppose that the following model is adopted y = X1β1 +
ε while the true model is y = X1β1 + X2β2 + ε. For both models, we
assume E(ε) = 0 and V (ε) = σ^2I. Figure out conditions under which
the least squares estimate we obtained is unbiased.

The output below is from a study that used multiple linear
regression analysis to link dependent variable y to x1
and x2.The sample consisted of 28 observations.
Coeff
Std Error
t Stat
P-value
Intercept
85
34.10
2.493
.0196
b1
-1.2
.88
-1.364
.1847
b2
3.5
1.12
3.125
Use the appropriate t tests to determine which, if any, of the
individual coefficients are statistically significant at the 5%
significance level. Report your conclusion. CHoose from A,B,C, and
D below
A)...

1. In a multiple
regression model, the following coefficients were obtained:
b0 = -10 b1
= 4.5 b2 = -6.0
a. Write the
equation of the estimated multiple regression model. (3 pts)
b Suppose a
sample of 25 observations produces this result, SSE = 480. What is
the estimated standard error of the estimate? (5 pts)
2. Consider the
following estimated sample regression equation:
Y = 12 + 6X1 -- 3 X2
Determine which of the following
statements are true,...

A multiple linear regression model based on a sample of 17 weeks
is developed to predict standby hours based on the total staff
present and remote hours. The SSR is 20,905.02 and the SSE is
25,434.29. (use 0.05 level of significance)
H0: B1 = B2 = 0
H1: At least one Bj does not equal 0, j = 1, 2
1. Calculate the test statistic.
Fstat= _____
2. Find the p-value.
p-value= _____
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