A1: the population regression funciton is linear in its parameters

A2: the values of the regressor x are fixed (not stochastic or random)

A3: the mean (conditional expectation) of the stochastic error term u_i is for any given value of x always zero (cancels out)

A4: for any given values of x, the conditioanl variance of the error term ui is a constant for any observation. (Homoscedasticity)

A5: For any x_i and x_j, we demand the correlation between the errors u_i and u_j is always zero (no autocorrelation)

A6: Regressor and error terms are not correlated (no covariance between error term and regressior)

A7: the number of observations (n) must be larger than the number of parameters (k)

A8: the variance of x must be positive finitie 

A9: must be correctly specified in all respects

A10: There can be no perfect multicollinearity between regressors

[image]

Must first calculate the

variance of the error term:

Var(u|x)=σ²

Standard Error of the Regression

(also called root mean squared error/RMSE)

σ=sqrt[∑u_i²/n-k-1]

• used as a "goodness of fit" measure 
• want it to be as small as possible (means the better the fit of the data)

• u_i's must have an expected value of zero
• must not be correlated to each other
• assume constant variance
• each error term must be normally distributed 

• To test the null hypothesis H₀: B_j=B_i

(looking for signficiant differences in the means)

• must calculate the t statistic to determine that

t ≡ (B_j-B_j^*)/se(B_j) ~ t_n-k-1

Stat programs test whether the regressor has any influence on y.  Thus, the H0=B_j=0 and the t stat= B_j/se(B_j)

Again, use t-test:

t ≡ B_j-B_l/se(B_j-B_l) ~ t_n-k-1

se(B_j-B_l)= sqrt[Var(B_j) + Var(B_l) - 2Cov(B_j,B_l)]

To question whether all the regressors taken together posses any explanatory power

H0: B₁=...=B_k=0

The null is rejected if calculated F> F from table

 [image]

Coefficient of Determination (aka=R²)

[image] 

Explained Sum of Squares (ESS)

ESS=∑(yhat-ybar)²

Residual Sum of Squares

RSS=∑u_i²

• If we had no regressors, the best estimate of y would be its mean

--> Therefore, the total variance of y is the 

Total Sum of Squares (TSS)

TSS=∑(y_i-yhat)²

TSS=ESS+RSS

What does R²=1 mean?

And R²=0?

R²=1 indicates that all observations lie on the regression line (perfect description/modelling of data)

R²=0 means that the regression model has no explanatory power with regards to y

R2 will never decrease if the number of regressors increases even if the newly added regressors have no real explanatory power

Adj. R² corrects for degrees of freedom (adding unneccessary regressors just to inflate R²)

Std. Error for regressor

t-stats for regressor and constant

p-values for regressor and constant

Confidence intervals 

F value for the model

Items asked to solve from STATA output

(Final Exam)

Model Mean Squared Error (MMS)

Number of df for residuals

R²

Root Mean Squared Errors (RMSE)

Confidence Intervals

Coefficients

With STATA output: SE of coefficient

(ß₁ in bivariate caes) 

ß₁/se(ß₁)

*easy because it is the two values

listed before it in the row  

reverse cumulative t distribution 

syntax:

1 - ttail(residual df, - t-stat) + ttail(residual df, + t-test)

CI: [ß₁ - tvalue*se(ß₁), ß₁ + tvalue*se(ß₁)]

tvalue retrieved from table

(usually 1.96 ~ 95%) 

What is meant by a "strong" effect?

What is meant by a "powerful" effect?

An effect is strong if the coefficient, compard to other coefficients, take on a large value (magnitude)

An effect is powerful/pronounced if the coefficient clearly exceeds its standard error

(that would be the ratio of coefficient to SE which equals the t-value)