Statistics Theory & Economics Theory Troubleshooting (Los Angeles MSA & San Francisco Bay Area MSA)

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This troubleshooting guide supports graduate students across the Los Angeles MSA & San Francisco Bay Area MSA, including UCLA, USC, UC Berkeley, UC Irvine, UC Davis, UC Santa Cruz, UC Riverside & all CSU campuses. It covers advanced Statistics Theory and Economics Theory topics found in graduate programs, including probability, distributions, estimation, hypothesis testing, microeconomic theory, macroeconomic theory, game theory & econometrics.

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Information Gain Section

This troubleshooting guide is designed to help you identify the exact theoretical gap causing confusion in your coursework. Each question isolates a specific concept—distribution properties, estimator behavior, equilibrium logic, or identification assumptions—so you can pinpoint the missing step in your reasoning. The goal is not to memorize formulas but to understand why the theory behaves the way it does, how assumptions shape results, and how to diagnose errors in your own derivations.

Statistics Theory & Economics Theory FAQ (50 Graduate-Level Questions)

1. What is the difference between a probability mass function and a probability density function?

A PMF assigns probability to discrete outcomes, while a PDF describes a continuous distribution where probabilities are computed as integrals over intervals.

2. How do I verify that a function is a valid PDF?

Check that it is nonnegative and integrates to 1 over its support.

3. How do I compute the expectation of a continuous random variable?

Use the integral:


E[X] = ∫ x f(x) dx

4. What is the law of total expectation?

It states that E[X] = E[E[X | Y]].

5. How do I compute variance using the shortcut formula?

Use:


Var(X) = E[X^2] - (E[X])^2

6. What is covariance and how is it interpreted?

Covariance measures joint variability; positive values indicate that variables move together.

7. How do I derive the moment generating function?


Compute the moment generating function (MGF) of X as:
Mₓ(t) = E[eᵗˣ]
whenever the expectation exists.

The MGF summarizes all moments of X. Differentiating Mₓ(t) and evaluating at
t = 0 gives the moments: Mₓ′(0) = E[X], Mₓ″(0) = E[X²], and so on.

8. What is the Cramér–Rao lower bound?

It provides a lower bound on the variance of unbiased estimators based on Fisher information.

9. How do I compute Fisher information?

Use:

The Fisher Information is defined as:
I(θ) = E[(∂/∂θ log f(X; θ))²]

It measures how much information the observable data X carries about the
parameter θ. A larger I(θ) implies more precision in estimating θ.
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10. What is the difference between unbiasedness and consistency?

Unbiasedness concerns finite-sample expectation; consistency concerns convergence as n → ∞.

11. How do I show an estimator is consistent?

Show convergence in probability to the true parameter.

12. What is asymptotic normality?

An estimator is asymptotically normal if its scaled error converges in distribution to a normal distribution.

13. How do I derive the MLE?

Write the log-likelihood, differentiate, set equal to zero & solve.

14. Why is the MLE asymptotically efficient?

Under regularity conditions, it achieves the Cramér–Rao lower bound asymptotically.

15. What is the likelihood ratio test?

It compares the maximized likelihood under the null and alternative hypotheses.

16. How do I compute a score function?

Differentiate the log-likelihood with respect to the parameter.

17. What is the delta method?

It approximates the distribution of a function of an estimator using Taylor expansion.

18. How do I compute a confidence interval using asymptotic theory?


Use the estimator ± z₍α⁄2₎ × standard error.

This gives a (1 − α) confidence interval by taking the point estimate and adding
and subtracting the critical value z₍α⁄2₎ multiplied by its standard error.

19. What is KL divergence?

It measures how one distribution diverges from another.

20. What is entropy in information theory?

Entropy measures uncertainty in a random variable.

21. What is the difference between pointwise and uniform convergence?

Uniform convergence requires convergence to hold uniformly over the domain.

22. How do I compute the characteristic function?

Use the characteristic function:
φₓ(t) = E[eⁱᵗˣ].

The characteristic function always exists (unlike the MGF) and uniquely determines the distribution of X. It is especially useful for proving limit theorems and working with sums of independent random variables.

23. What is Slutsky’s theorem?

It describes how convergent sequences behave under algebraic operations.

24. What is the continuous mapping theorem?

It states that continuous functions preserve convergence in distribution.

25. How do I prove the central limit theorem for i.i.d. variables?

Use characteristic functions and Lindeberg–Feller conditions.

26. What is a competitive equilibrium?

It is a price–allocation pair where supply equals demand and agents optimize.

27. How do I derive a consumer’s Marshallian demand?

Solve the utility maximization problem subject to the budget constraint.

28. What is Hicksian demand?

It minimizes expenditure subject to achieving a target utility level.

29. What is the Slutsky equation?

It decomposes price effects into substitution and income effects.

30. How do I compute a firm’s cost function?

Minimize cost subject to producing a given output level.

31. What is a Nash equilibrium?

A strategy profile where no player benefits from unilateral deviation.

32. How do I solve a normal-form game?

Identify best responses and find mutual best-response strategies.

33. What is subgame perfect equilibrium?

An equilibrium that prescribes optimal strategies in every subgame.

34. What is the Euler equation in macroeconomics?

It characterizes optimal intertemporal consumption.

35. How do I derive the IS curve?

Combine goods market equilibrium with investment–interest relationships.

36. What is the Phillips curve?

It relates inflation to unemployment or output gaps.

37. What is the Solow growth model?

It describes long-run output based on capital accumulation & technology.

38. What is the difference between static and dynamic optimization?

Static optimization solves one-period problems; dynamic optimization solves multi-period problems.

39. What is the Lagrangian for a constrained optimization problem?

It incorporates constraints using multipliers.

40. How do I derive first-order conditions?

Differentiate the Lagrangian with respect to choice variables and set equal to zero.

41. What is identification in econometrics?

Identification means the parameter is uniquely determined by the distribution of observables.

42. What is endogeneity?

Endogeneity occurs when regressors correlate with the error term.

43. What is the instrumental variables estimator?

It uses instruments to isolate exogenous variation in regressors.

44. How do I compute the IV estimator?

Use:

βᴵⱽ = (Z′X)⁻¹ Z′Y
Here Z contains the instruments, X the endogenous regressors, and Y the outcome.
This expression gives the two‑stage least squares (2SLS) estimator in its compact matrix representation.

45. What is the Gauss–Markov theorem?

It states that OLS is BLUE under classical assumptions.

46. What is heteroskedasticity?

It occurs when error variance is not constant across observations.

47. What is autocorrelation?

It occurs when errors are correlated across time.

48. What is maximum likelihood estimation in econometrics?

It chooses parameters that maximize the likelihood of observing the data.

49. What is the difference between structural and reduced-form models?

Structural models impose economic theory; reduced-form models describe statistical relationships.

50. What is the difference between stationarity and nonstationarity?

Stationary processes have constant mean and variance over time.

FAQ — Economics Tutoring (15 Questions)

1. What is the first‑order condition for utility maximization?

Set marginal rate of substitution equal to the price ratio:


MRS = MU₁ / MU₂ = p₁ / p₂

2. What is Hicksian demand?

Demand that minimizes expenditure for a target utility level.

3. What is the Slutsky equation?

It decomposes price effects:


∂xᵢ/∂pⱼ = ∂hᵢ/∂pⱼ − xⱼ ∂xᵢ/∂I

4. What is the Euler equation?

The optimal consumption condition in dynamic models:


u'(cₜ) = β (1+r) u'(cₜ₊₁)

5. What is a steady state?

A point where variables stop changing:


k* = s f(k*) − δk*

6. What is Nash equilibrium?

A strategy profile where no player can gain by deviating.

7. What is subgame perfection?

A refinement requiring Nash equilibrium in every subgame.

8. What is the difference between FE and RE?

FE removes time‑invariant heterogeneity; RE assumes it is uncorrelated with regressors.

9. What is the IV estimator?

The ratio of reduced‑form to first‑stage coefficients:


β_IV = Cov(Z,Y) / Cov(Z,X)

10. What is the difference between ATE and LATE?

ATE is the average effect; LATE applies to compliers in IV settings.

11. What is the Bellman equation?

The recursive representation of dynamic optimization:


V(k) = max_c { u(c) + β V(k') }

12. What is the Solow model’s key prediction?

Economies converge to a steady state determined by savings, depreciation & technology.

13. What is a best response?

A strategy that maximizes payoff given the opponent’s strategy.

14. What is the difference between short‑run and long‑run cost?

Short‑run has fixed inputs; long‑run allows all inputs to vary.

15. What is the likelihood function?

The probability of observing the data given parameters:


L(θ) = Π f(yᵢ | θ)

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