ECON 709 Metrics Reference

Flashcards created for review and study purposes based on ECON 709 Metrics reference notes.

What is the formula for the Autocorrelation Function in AR(1) processes?

ρk = ϕk

What does the Partial Autocorrelation at lag k identify?

It identifies the AR order.

How do you calculate Effective Degrees of Freedom in a model?

dfeff = tr(S), where S is the smoothing matrix.

What is the variance of the sample mean when variables are independent?

Var(X̄) = σ²/n.

What is the T-statistic for the mean?

t = (X̄ − µ₀) / (s / √n).

How is Balanced Accuracy calculated?

Balanced Accuracy = (TPR + TNR) / 2.

What does the term Bias of an estimator refer to?

Bias(θ̂) = E[θ̂] − θ.

What is the formula for Quantile Loss?

Lτ(r) = τr, r ≥ 0; Lτ(r) = (τ − 1)r, r < 0.

What is the Bonferroni Inequality for two sets?

P(A ∪ B) ≥ P(A) + P(B) − 1.

What does RMSE stand for?

Root Mean Squared Error.

How is the Standard Error of the mean computed?

SE(X̄) = σ/√n.

What are Type I and Type II Errors in hypothesis testing?

α = P(Reject H₀ | H₀ true), β = P(Fail to reject H₀ | H₁ true).

What defines Conditional Independence?

P(A ∩ B | C) = P(A | C) P(B | C).

What does Conditional Probability represent?

P(A | B) = P(A ∩ B) / P(B), P(B) > 0.

What formula represents Conditional Variance?

Var(Y | X) = E[(Y − E[Y | X])² | X].

What criterion does the Confidence Interval - Inversion satisfy?

H₀ not rejected iff θ₀ ∈ [θ̂ ± t₁−α/2, se(θ̂)].

What components are in a Confusion Matrix for binary classification?

TP, FP, TN, FN.

What is the definition of Consistency of an Estimator?

θ̂ₙ p−→ θ.

What does the Continuous Mapping Extension state?

If (Xₙ d−→ X) and g is continuous, then (g(Xₙ) d−→ g(X)).

What is the Continuous Mapping Theorem?

If (Xₙ p−→ X) and g is continuous, then (g(Xₙ) p−→ g(X)).

What does Joint Convergence in Probability imply?

If (Xₙ p−→ X) and (Yₙ p−→ Y), then ((Xₙ, Yₙ) p−→ (X, Y)).

How is the Coefficient of Determination ($R^2$) calculated?

R² = 1 − (∑(yᵢ − ȳ)² / ∑(yᵢ − ŷᵢ)²).

What defines the Joint Cumulative Distribution Function (CDF)?

F_X,Y (x,y) = P(X ≤ x, Y ≤ y).

What does the Delta Method (Univariate) signify?

If (√n(θ̂ − θ) d−→ N(0, σ²)), then (√n(g(θ̂) − g(θ)) d−→ N(0, [g'(θ)]²σ²).

What is Deviance in Generalized Linear Models?

D = 2(ℓsaturated − ℓmodel).

How is the Dice Coefficient defined?

2TP / (2TP + FP + FN).

What is the formula for the Expectation of the sample mean?

E[X̄] = µ.

How is the Explained Sum of Squares (ESS) calculated?

ESS = ∑(ŷᵢ − ȳ)².

What does the Precision–Recall Curve represent?

Area Under Precision–Recall (AUPRC) = ∫ Precision(Recall) d Recall.

What is the MA(q) Process Variance formula?

σ²x = σ²ε ∑_(i=0)ⁱ θ²ᵢ.

How is the Arithmetic Mean computed?

x̄ = (1/n) ∑(xᵢ).

What does Hinge Loss measure in SVM?

L = (1/n) ∑ max(0, 1 − yᵢ f(xᵢ)).

What characterizes an IID Random Sample?

(X₁, …, Xₙ iid∼ (µ, σ²)).

What does the Inclusion–Exclusion Principle express?

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

What is the Interquartile Range (IQR)?

IQR = q₀.75 − q₀.25.

What does the Jaccard Index (IoU) measure?

TP / (TP + FP + FN).

What does the Large-Sample Normal Approximation state?

If (√n(θ̂ − θ) d−→ N(0, V)), then (z = (θ̂ − θ₀) / (V/n) ∼ N(0, 1)).

What is the Lasso Objective in regression?

min β (1/2n) || y − Xβ ||² + λ||β||₁.

What does the Law of Iterated Expectations (LIE) indicate?

E[E[Y | X]] = E[Y].

What does the Law of Total Probability express?

P(A) = ∑ P(A | Bᵢ) P(Bᵢ).

How is the Law of Total Variance formulated?

Var(Y) = E[Var(Y | X)] + Var(E[Y | X]).

What is the significance of Levy’s Continuity Theorem?

Xₙ d−→ X ⇐⇒ MXₙ(t) → MX(t) for t in a neighborhood of 0.

What does the Likelihood Ratio Test calculate?

LR = 2ℓ(θ̂) − ℓ(θ₀) d−→ χ²ₓ.

How is Log Loss or Binary Cross-Entropy defined?

−(1/n) ∑ [yᵢ log(ŷᵢ) + (1 − yᵢ) log(1 − ŷᵢ)].

What defines the M-th Central Moment?

µₘ = E[(X − E[X])ₘ].

What is the formula for the M-th Raw Moment?

mₘ = E[Xᵐ].

How is the Variance of Sum derived?

Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y).

What does the Matthews Correlation Coefficient (MCC) evaluate?

√ (TP·TN − FP·FN) / ((TP + FP)(TP + FN)(TN + FP)(TN + FN)).

How is the Mean Absolute Error (MAE) defined?

MAE = (1/n) ∑ |yᵢ − ŷᵢ|.

What does Mean Absolute Scaled Error (MASE) compare?

MASE = MAE / (1/(n−1) ∑ |yᵢ − yᵢ₋₁|).

How is Mean Bias Error (MBE) calculated?

MBE = (1/n) ∑ (ŷᵢ − yᵢ).

What is the Mean Squared Error (MSE) formula?

MSE(θ̂) = Var(θ̂) + Bias(θ̂)².

What is the Mean of a Random Variable?

µ = E[X].

How is the Median Absolute Error defined?

median|yᵢ − ŷᵢ|.

What characterizes the Moment Generating Function (MGF)?

MX(t) = E[e^(tX)], MX+Y(t) = MX(t)MY(t).

What are the Null and Alternative Hypotheses?

H₀ : θ = θ₀ vs. H₁ : θ ≠ θ₀.

How are OLS Normal Equations expressed?

(XᵀX)β̂ = Xᵀy.

What describes the One-Sample t-Test?

t = (X̄ − µ₀) / (s / √n).

What is the One-Sample z-Test formula?

z = (X̄ − µ₀) / (σ / √n).

What is the Right-Tail P-Value?

p = 1 − FT(tobs).

What does Pairwise Independence show?

P(Aᵢ ∩ Aⱼ) = P(Aᵢ)P(Aⱼ) ∀i ≠ j.

What is the Power of a Test formula?

Power = 1 − β.

What is Precision in the context of predictive model evaluation?

Precision = TP / (TP + FP).

What does PDF stand for?

Probability Density Function.

How is the Probability Mass Function (PMF) defined?

p_X(x) = P(X = x).

What is the Quantile / Quartile Definition?

qα = inf x : FX(x) ≥ α.

How is Generalization Error decomposed?

E[(y − f̂(x))²] = (E[f̂(x)] − f(x))² + Var(f̂(x)) + σ².

What does the ROC Curve illustrate?

Trade-off between TPR and FPR across thresholds.

How is Recall or Sensitivity calculated?

Recall = TP / (TP + FN).

What does the RSS (Residual Sum of Squares) measure?

Total unexplained variation after fitting the model.

What is the Ridge Solution in linear regression?

β̂_ridge = (XᵀX + λI)⁻¹Xᵀy.

What does the Score (Lagrange Multiplier) Test evaluate?

LM = S(θ̂₀)ᵀ[I(θ̂₀)⁻¹]S(θ̂₀).

What is the Seasonal Naive Forecast?

ŷt = y{t-s}.

What does Spearman Rank Correlation measure?

Monotonic association via ranks.

How is Standard Deviation calculated?

σ = √Var(X).

What formula is used for the probability of an interval for continuous RV?

P(a ≤ X ≤ b) = ∫[b,a] f_X(x) dx.

What does the Strong Law of Large Numbers state?

X̄_n a.s. −−→ µ.

What does Symmetric Mean Absolute Percentage Error (sMAPE) account for?

sMAPE = 100/n ∑ |yᵢ − ŷᵢ| / ((|yᵢ| + |ŷᵢ|)/2).

How is the T-Statistic for the Mean calculated?

t = (X̄ − µ₀) / (s / √n).

What happens to the T-Statistic limit as sample size increases?

tn−1 → z as n → ∞.

What is the expected value of a function of RV?

E[g(X)] = ∫ g(x)p_X(x)dx.

What is the General Form of a Test Statistic?

T = (Estimator - Hypothesized Value) / Standard Error.

How is Total Sum of Squares (TSS) expressed?

TSS = ∑(yᵢ − ȳ)².

How is the Unbiased sample variance defined?

s² = (1 / (n − 1)) ∑(Xᵢ − X̄)².

What defines the Variance Inflation Factor (VIF)?

VIFj = 1 / (1 − R²j).

What does the Variance Identity state?

Var(X) = E[X²] − (E[X])².

How is the Variance of an Affine Transformation calculated?

Var(aX + b) = a²Var(X).

What does the Variance–Covariance Matrix represent?

(σ = [Var(X), Cov(X, Y); Cov(Y, X), Var(Y)]).

What does the Wald Test evaluate?

W = (g(θ̂))ᵀ[Var(g(θ̂))]⁻¹g(θ̂).

What does the Weak Law of Large Numbers (Khinchin) state?

If(Xi)i.i.d. with (E[Xi] = µ), then (X̄_n p−→ µ).

How is the Variance of the Sample Mean represented?

Var(X̄) = σ²/n.

How is the Mean Absolute Percentage Error (MAPE) defined?

MAPE = (100/n) ∑ |(yᵢ − ŷᵢ) / yᵢ|.

What relationship exists between PMF and CDF for discrete distributions?

FX(x) = ∑{t≤x} p_X(t).

What formula represents Sample Variance?

s² = (1/(n−1)) ∑(xᵢ − x̄)².

What does the Harmonic Mean evaluate?

H = 1 / (1/n ∑(1/xᵢ)).

What does Jensen’s inequality imply about convex transforms?

If (g) is convex, then g(E[X]) ≤ E[g(X)].

What does the formula
ho_k = \phi^k describe for an AR(1) process?

It describes how the autocorrelation of an AR(1) process decays exponentially with lag k.

What is the purpose of the t-statistic formula t = (\bar{X} - \mu_0) / (s / \sqrt{n})?

It is used to test hypotheses about the population mean \mu when the population standard deviation \sigma is unknown.

What does the Bias of an estimator, Bias(\hat{\theta}) = E[\hat{\theta}] - \theta, quantify?

It quantifies the difference between the expected value of an estimator and the true value of the parameter being estimated.

What is the practical application of the Bonferroni Inequality P(A \cup B) \geq P(A) + P(B) - 1?

It provides a lower bound for the probability of the union of two events, useful in scenarios like multiple comparisons.