Econometrics

total sum of squares = sum of (yi-ybar) squared

explained sum of squares = sum of (yi hat - yi bar) squared

residual sum of squares / sum of squared residuals = sum of (ui hat) squared

SSE + SSR

SSE/SST or 1-SSR/SST, interpreted as the proportion of the sample variation in yi that is explained by the OLS regression line

sample population

sample population mean

fitted equation variable

1. linear in parameters 2. random sample of n observations 3. no perfect collinearity 4. zero conditional mean

the procedure by which the OLS estimates are obtained is unbiased when we view the procedure applied across random samples. estimator Bhat is unbiased estimator of B if the mean of B hats sampling distribution is equal to B.

Too many independent variables; does not affect unbiasedness but does affect variances of estimators

E(B tilda) = E (B hat + B hat * delta tilda (slope from simple regression) = E(B hat) + E (B2 hat)*deta tilda = B + B2*delta tilda. Bias in (B tilda) = E(B tilda) - B=B2*delta tilda. If B2 or delta tilda = 0, unbiased. NOTE TABLE OF BIAS

simplifies formulas; error u has the same variance given any values of the explanatory (x) variables. in other words, Var(u|x...xk)= sigma squared

error u has different variance given different values of explanatory variables

Gauss-Markov assumptions

= Var(B hat) or Variance of OLS multi reg

high but not perfect correlation between two or more independent variables; is fine esp when found between other explanatory variables that are not variable of interest

n-k-1 where n is observations and k is independent variables: count number of parameters including intercept, subtract amount from number of observations

se(Bj hat) = root (sigma hat / [SSTj * (1-R squared j )])

best linear unbiased estimator --> OLS

Assumption that unobserved population error u is independent of the explanatory variables AND is normally distributed with zero mean and variance sigma squared: u ~ Normal (0, sigma squared)

All 5 Gauss-Markov + Normality

If true: once X1..Xj-1 have been accounted for, Xj has not effect on expected value of y. Presumed true unless rejected (rejected or not rejected)

chosen value that makes area in each tail of the t distribution equal to total significance level. Ex for significance level 5%, c = value that makes area equal to 5% (one-tailed) or 2.5% (two tailed). 1.96 is critical value for N>120 two tailed test at 5% confidence **

if absolute value of t stat of B hat is greater than critical value c, reject null hypothesis

= Beta hat / se(Beta hat) --> Beta hat / root (sigma hat / ((yi- yi bar) squared)* [(1-Rj squared)])

if null hypothesis is rejected, we cannot say ind var has no effect, so we say "xj is statistically significant at the (significance)% level". If not rejected, "xj is statistically insignificant at the (significance)% level"

t = (estimate- hypothesized value) / standard error --> if t>c, reject Ho in favor of H1

Relationship between dependent and independent variables is linear

Independent variables are not perfectly correlated with each other

Observations are independently and identically distributed (i.i.d)

The error term has a mean of zero given any value of the independent variables

Measures how far X is from E(X0 or how far away it is from it's mean. Var(x) = E[(X-Xbar)^2] or = SUM [xi-xbar]^2 / N-1

Measures amount of linear dependence between two random variables. Positive: they move together. Negative: they go opposite directions. Cov(X,Y)=E[(X−EX)(Y−EY)]=E[XY]−(EX)(EY). OR SUM [xi-xbar]*[yi-ybar] / N-1

Assign magnitude of perfectly positive, perfectly negative, or uncorrelated relationship. Corr(x,y) = cov(x,y)/sd(x)sd(y)

Estimated Slope. OLS, defined as SUM (yi-ybar)(xi-xbar) / SUM (xi-xbar) or Cov(x,y) / Var(x)

square root of variance

Small p-values evidence against the null, large p-values don't provide evidence against the null. In simpler terms, it tells you how likely it is to get the observed results if there is no real effect or relationship in the population. Want a small p value.

Range of likely values for the population parameters. Bhat (estimate) +/- c*se(Bhatj). Confidence interval for where unknown B lies --> if 95%, then Bj lies on interval (Blower, Bupper) for 95% of the samples.

Used for testing joint hypothesis of set of parameters: null hypothesis is that a set of variables has no effect on y once another set of variables has been controlled for.
F = (SSR restricted - SSR unrestricted)/q / SSR unrestricted/(n-k-1). q = n-k-1 restricted - n-k-1 unrestricted.

concluding null hypothesis is false when it is true --> false positive

concluding null hypothesis is true when it is false

1. define: E[Bhat]=B
2. Bhat matrix form = (X'X)^(-1) * X'Y --> plus this in to Bhat
3. DGP is Y = XB + Epsilon --> replace Y with XB + Epsilon
4. Simplify to B + E[(X'X)^)-1)X'Epsilon

1. Scale - some entities are larger than others 2. Opportunities are different across groups 3. Diversity in cross-sectional sample

It does not bias the estimators, but the standard errors will be wrong, and we don't know whether the standard errors are too big or too small. This would affect t-stats and f-tests making our inferences incorrect.

1) Plot residuals and use eyeball test 2. Goldfeld-quandt- basically separate data where the errors vary and run regressions to reject homo 3. Breusch-pagan

Use transformation by multiplying all variables by 1oot(x) or Huber White Estimator which adjusts standard errors

Adjusts standard errors to account for within-unit heteroskedasticity. Allows for between-unit differences is variance and within-unit correlation of errors. *always makes SE larger

Yit = a + BXit + Eit
Only best when unit-specific intercepts are zero. No unobserved hetero and errors are independent across each unit-time observation.

Yit = ai + BXit + Eit
Capture the relationship between the dependent variable and time-varying independent variables within each entity (e.g., individual, firm, country) over time. This means that variables that change within each entity over time are included in the estimation process. Time-Invariant Characteristics: Time-invariant characteristics, such as individual talent or firm-specific attributes, are absorbed by the entity-specific intercepts in fixed effects models. Is best when loss of precision preferred to bias & unmeasured heteroskedasticity is correlated with X.

Yit = a + BXit + Uit (U is composite error of ai + Eit)
Weighted avarage of within and between models, using both within and between entity variation. Unit-specific intercepts -each Ai in the Uit represents the deviation of an individual from the overall population mean due to unobserved factors, as part of larger distribution.By treating ai as random, the random effects model allows for inferences to be generalized to a broader population rather than being limited to the specific sample used in the analysis. This is particularly useful in hierarchical or multilevel data structures where entities (e.g., individuals, firms, schools) are sampled from a larger population. Is best when unmeasured heterogeneity is uncorrelated with X.

Expensive in terms of degrees of freedom. The precision of the coefficient estimates decreases as degrees of freedom decrease. The statistical power to detect true effects diminishes because the standard errors of the estimates increase.

Used to compare RE and FE coefficient estimates. If sufficiently different, use FE.

variation in the data that occurs within each entity over time -- captures how the variables change for the same entity across different time periods.

variation in the data that occurs between different entitites -- captures differences between entities that are constant over time.