Formula sheet

a question asks you to justify why econometrics is not simply 'applied statistics' or 'mathematical economics.'

you are asked to summarise, in one line, what econometric research is ultimately for.

an exam asks you to list, or apply, the standard stages of an empirical econometric study — walk through each with the running example below.

$Q$

quantity demanded (dependent variable)

$P$

the good's own price (explanatory)

$I$

consumer income (explanatory)

$P_s$

price of a substitute (explanatory)

$P_c$

price of a complement (explanatory)

$Z$

other factors such as tastes and preferences (explanatory)

step 2 needs the model's variables named before a functional form is selected.

$Y$

consumption

$X$

income (here, GDP)

$\beta_0$

the intercept (constant term)

$\beta_1$

the slope, the marginal propensity to consume (MPC), between 0 and 1

the mathematical (deterministic) stage of the model, step 2, before a disturbance term is added.

$u$

the disturbance (error) term, a random variable capturing everything the deterministic line leaves out

the model is now stochastic rather than exact — the standard starting point for every regression in this course.

$\widehat{MPC}$

the estimated marginal propensity to consume (slope from the fitted line)

a fitted consumption function's slope is used to estimate the impact of a change in government spending on national income — a direct step-8 application.

you are given a dataset's description and need to classify it — a very common short-answer question.

a question asks you to explain why a regression result does not, by itself, prove causation.

$E(Y \mid X_i)$

the conditional mean of Y for a given value X_i

$\beta_1, \beta_2$

the true, unknown population parameters

you need to state the theoretical object regression is trying to uncover — the true relationship in the whole population, not just in one sample.

$u_i$

the disturbance (error) term for observation i, capturing everything that pulls Y_i away from its conditional mean

you write down the full population model, ready to be estimated from a sample — this equation is the starting point of every two-variable regression in this course.

$\hat\beta_1, \hat\beta_2$

estimated parameters (written b_1, b_2 on the slides)

$\hat Y_i$

the fitted (predicted) value

$e_i$

the residual, the SAMPLE counterpart of u_i

you are working with actual estimated numbers from a specific dataset, as opposed to the unobservable, purely theoretical PRF.

a model has a squared or otherwise nonlinear term in X, and you need to decide whether ordinary least squares can still be used.

$SSR$

sum of squared residuals

$b_1, b_2$

the candidate intercept and slope being chosen

you need to explain what OLS is actually optimizing, before deriving the formulas for b1 and b2.

$\bar X, \bar Y$

the sample means of X and Y

$S_{xy}$

sum of products of deviations from the means

$S_{xx}$

sum of squared deviations of X from its mean

you need to compute the OLS intercept and slope by hand from raw (X,Y) data — the single most-used formula pair in this course.

$x_i, y_i$

deviations from the means

$\sum X_iY_i, \sum X_i^2$

sums of the raw values

$n$

sample size

you are given raw sums (the second form) or a deviation table (the first form); both give the same number.

$TSS$

total sum of squares

$ESS$

explained sum of squares

$SSR$

residual (unexplained) sum of squares

you need to break down the total variation in Y into the part the regression line explains and the part it leaves unexplained.

$R^2$

the proportion of the total variation in Y explained by the regression, 0 \leq R^2 \leq 1

a question asks how well the fitted line explains the data — the standard, single-number goodness-of-fit measure for a two-variable regression.

$\sigma^2$

variance of the disturbance u

$x_i$

X_i - \bar X

$\sum x_i^2$

total variation of X around its mean

you need the precision of the slope estimate, or the SE for a t test or a confidence interval.

$\sum X_i^2$

sum of the squared raw X values (capital X)

$n$

sample size

you need the precision of the intercept. Careful: the numerator has capital X, the denominator small x.

$RSS$

\sum e_i^2 = residual sum of squares

$n - 2$

degrees of freedom (two parameters estimated)

$\hat\sigma$

standard error of the regression

you need a number for sigma-squared before computing any variance or standard error. EViews prints sigma-hat as 'S.E. of regression'.

$\bar X$

sample mean of X

you are asked how b1 and b2 move together: with a positive mean of X they are negatively related (overestimate the slope, underestimate the intercept).

$r^2$

share of the total variation in Y explained by the regression

$y_i$

Y_i - \bar Y

you need the fit of a two-variable regression; the last two forms work straight from the deviation sums without residuals.

$r$

sample correlation between X and Y, with the same sign as b_2

you are asked for the correlation between X and Y after a two-variable regression.

$1 - \alpha$

confidence coefficient (e.g. 0.95)

$\alpha$

level of significance (e.g. 0.05)

you need to state formally what a confidence interval promises.

$b_2$

sample estimate of the slope

$\beta_2$

true population slope

$SE(b_2)$

standard error of b_2

$n-2$

degrees of freedom

you need to turn an estimate and its standard error into a quantity with a known distribution, the starting point for both intervals and tests.

$b_2$

sample value of the slope

$t_{\alpha/2}$

two-sided critical value from the t table with n-2 df

$SE(b_2)$

standard error of b_2

a question asks for an interval estimate of the slope, or 'in which interval will Y change when X rises by one unit'.

$b_1$

sample value of the intercept

$SE(b_1)$

standard error of b_1

a question asks for an interval estimate of the intercept; the recipe is identical, only the coefficient and its standard error change.

$c$

hypothesised slope under H0

$b_2$

sample estimate

$SE(b_2)$

its standard error

you test any claimed value of a coefficient. With c = 0 it reduces to coefficient / standard error, the t-Statistic column in EViews.

$R^2$

coefficient of determination: the share of the total change in Y explained by the regression

you need to say how well the regression explains Y as a whole.

$ESS/1$

mean square of the regression

$RSS/(n-2)$

mean square of the residuals, the estimated error variance

$F_{1,n-2}$

F distribution with 1 and n-2 df

you test whether the regression explains Y at all (H0: beta2 = 0) from the ANOVA table.

$JB$

Jarque-Bera test statistic

$n$

sample size

$S$

skewness coefficient

$K$

kurtosis coefficient

you need to test whether the regression residuals follow a normal distribution.

$\operatorname{var}(\hat{Y}_0)$

variance of predicted mean Y

$\sigma^2$

disturbance variance

$n$

sample size

$X_0$

chosen regressor value

$\bar{X}$

sample mean of X

$\sum x_i^2$

sum of squared deviations of X

you need the sampling variance of the estimated conditional mean E(Y|X0).

$\text{Elasticity}$

percentage responsiveness of Y to X

$\frac{dY}{dX}$

slope of the function

$X$

regressor value

$Y$

regressand value

you need to calculate the percentage response of Y to a 1% change in X.