EKHM65 Econometrics I: Random Variables, Estimators, Hypothesis Testing, and Regression

Comprehensive practice flashcards covering random variables, probability moments, estimator properties, the Law of Large Numbers, the Central Limit Theorem, hypothesis testing with the t-test, covariance, correlation, and the linear regression model.

In econometrics, what is the definition of a random variable prior to sampling?

A random variable is defined as a variable whose value is uncertain until it is observed; after sampling, it is realized as a single fixed number.

How are the third and fourth moments of a probability distribution (skewness and kurtosis) defined and benchmarked?

Skewness is defined as $\frac{E[(Y - \mu)^3]}{\sigma^3}$ ($0$ for symmetric, $> 0$ for a right tail, and $< 0$ for a left tail); kurtosis is defined as $\frac{E[(Y - \mu)^4]}{\sigma^4}$ ($3$ for a normal distribution, $> 3$ for leptokurtic / heavy tails, and $< 3$ for platykurtic / light tails).

What condition must an estimator satisfy to be considered unbiased?

An estimator $\hat{\mu}_Y$ is unbiased if the mean of its sampling distribution equals the true population parameter being estimated, such that $E(\hat{\mu}_Y) = \mu_Y$.

What are the two formal criteria that define an idealized random sample as independent and identically distributed (i.i.d.)?

Identically distributed means each observation $Y_i$ is drawn from the same population distribution, and independently distributed means the realization of one observation provides no information about the realization of another.

Why is the probability of a continuous random variable taking any exact specific value equal to zero ($P(X = x) = 0$)?

Because a continuous random variable can take infinitely many values in an interval; probabilities are therefore defined over ranges ($P(a < X < b)$) and measured by the area under the density curve.

How is relative efficiency defined when comparing two unbiased estimators?

Among unbiased estimators, the one with the smaller variance is more efficient; formally, if $\text{Var}(\hat{\theta}_1) < \text{Var}(\hat{\theta}_2)$, then $\hat{\theta}_1$ is more efficient.

What does the Law of Large Numbers (LLN) state regarding the sample average?

Provided that $Y_1, \dots, Y_n$ are i.i.d. and $\text{Var}(Y_i)$ is finite, as the sample size $n$ increases, the sample average becomes increasingly concentrated around the population mean.

How does the concept of consistency differ from the concept of unbiasedness?

Consistency describes the behavior of an estimator as sample size approaches infinity ($n \rightarrow \infty$), ensuring the probability of being within any margin of error approaches $1$, whereas unbiasedness describes the expected value of an estimator across repeated samples of a fixed sample size.

In the Swedish age simulation across 1,000 samples, what were the standard deviations of the sampling distribution of the sample average for $n = 30$, $n = 100$, and $n = 1{,}000$?

The standard deviation of the sampling distribution was $2.63$ for $n = 30$, $1.45$ for $n = 100$, and $0.49$ for $n = 1{,}000$.

What does the Central Limit Theorem (CLT) state regarding the standardized sample average?

Provided that $Y_1, \dots, Y_n$ are i.i.d. and $\text{Var}(Y_i)$ is finite, when $n$ is large, the sampling distribution of the standardized sample average is approximately standard normal ($N(0, 1)$), even if the underlying observations are not normally distributed.

When an observed sample average $\bar{Y}$ differs from a hypothesized mean $\mu_0$, what are the two mutually exclusive explanations?

Either the hypothesis is wrong ($\mu_Y \ne \mu_0$), meaning the true mean differs from the hypothesized value, or the hypothesis is correct but sampling noise occurred ($\mu_Y = \mu_0$, yet $\bar{Y} \ne \mu_0$).

What is the key difference between the sample standard deviation ($s_Y$) and the standard error of the sample average ($SE(\bar{Y})$)?

The standard deviation describes how individual observations differ from one another, whereas the standard error ($SE(\bar{Y}) = \frac{s_Y}{\sqrt{n}}$) describes how much the sample average varies from sample to sample.

What is the formula for the t-statistic when testing the population mean with an unknown population standard deviation?

$t = \frac{\bar{Y} - \mu_0}{SE(\bar{Y})}$, where $SE(\bar{Y}) = \frac{s_Y}{\sqrt{n}}$, and the statistic follows a t-distribution with $n - 1$ degrees of freedom.

In the test of $H_0: \mu_Y = 50$ versus $H_1: \mu_Y \ne 50$ with $n = 30$, $\bar{Y} = 47.2$, and $SE(\bar{Y}) = 2.91$, what was the calculated t-statistic and the decision at the $5\%$ level?

The t-statistic was $t = \frac{47.2 - 50}{2.91} = -0.96$; because $-0.96$ falls between the critical values of $\pm 2.05$ ($29$ degrees of freedom), the decision was to fail to reject $H_0$.

Why did testing $H_0: \mu_Y = 50$ with a sample size of $n = 1{,}000$ lead to rejecting the null hypothesis when $n = 30$ failed to reject it?

The larger sample size reduced the standard error from $2.91$ to $0.48$ via the $\sqrt{n}$ term, which produced a t-statistic of $-6.88$ that far exceeded the critical cutoff of $\pm 1.96$.

How is the sample covariance ($s_{XY}$) calculated and why is its magnitude alone difficult to interpret?

$s_{XY} = \frac{1}{n - 1} \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})$; while its sign indicates the direction of linear co-movement, its magnitude depends directly on the units of measurement of $X$ and $Y$.

Why can two variables have a correlation coefficient of $r_{XY} = 0$ even when there is a strong mathematical relationship between them?

Correlation measures only linear association; if the underlying relationship is non-linear (such as an inverted-U quadratic curve), positive and negative deviations cancel each other out in the summation, yielding a correlation of zero.

What limitation of correlation is demonstrated when straight lines of different steepness all have a correlation of $1$?

Correlation indicates how tightly data points hug a line, but it cannot measure how steep the line is or how much $Y$ changes per unit change in $X$, which is instead provided by the regression slope.

In the linear regression model $Y_i = \beta_0 + \beta_1 X_i + u_i$, what do the terms $\beta_1$, $\beta_0$, and $u_i$ represent?

$\beta_1$ is the slope (change in the conditional mean $E(Y|X)$ per one-unit increase in $X$), $\beta_0$ is the intercept (conditional mean of $Y$ when $X = 0$), and $u_i = Y_i - E(Y_i|X_i)$ is the error term.