Pytania - II II QUANTITATIVE ANALYSIS
Q38: Describe and interpret a confidence interval
A: CI estimates result in a range of values given probability 1-alpha alpha - lvl of significance E.g. estimate that population mean lies between 15 and 25 with a 95% degree of confidence or at 5% lvl of significance Construction: point estimate +/- (reliability factor * standard error) For normal dist with known var -> mean +/- z(alpha/2) * sigma/sqrt(n) Two ways of interpretation: Probabilistic: after repeatedly taking samples exam candidates and constructing confidence intervals for each sample's mean, 99% of resulting CI will, in the long run include the population mean Practical: we are 99% confident that population mean score lies between ... and ... for candidates in this population Normal dist with uknown variance -> mean +/- t(alpha/2)*s/sqrt(n) Nonnormal dist but population variance known -> z-statistic can be used, n large! (n>=30) Nonnormal dist with unknown variance -> t-statistic can be used, n large! (n>=30) (acceptable z-stat, but t-stat more conservative Nonnormal dist known/unknown variance and n<30 - no way to create a confidence interval
Q72: Describe examples of omitted variable bias
A: 2 conditions can occur: Omitted variable is a determinant of dependent var OV correlated with at least one independent var
Q56: Describe the effect of heteroskedasticity on regression analysis (3)
A: 3 major effects: *Standard errors unreliable *Coefficient estimates aren't affected *If standard error too small, but b(j) unaffected - t-statistics too larche and H0 rejected falsely tooo often.
Q58: Describe properties of OLS estimators according to Gauss-Markov theorem (4)
A: 4 key properties: *OLS estimated coeffs have min variance compared to other methods *OLS coeff based on linear functions *OLS coeffs unbiased - in repeated sampling the averages will be distributed around true population parameters *OLS estimate of variance of errors unbiased Acronim of these properties - BLUE - best linear unbiased stimators
Q90: Describe appilcation of AR and ARMA processes
A: AR - if autocorrelations decay gradually, don't cut off abruptly (AR stays, MA out) ARMA - if both: gradual decay and abrupt cut off Another way 0 test various models using regression results
Q86: Describe the autoregressive process
A: AR is inverted MA More useful - expresses current observables in terms of past observables. AR(1) has too mean 0 and variance constant -> y(t) = phi*y(t-1) + e(t), phi - coeff for lagged observ of var, e(t) - current random white noise shock Predictive ability ~ stationarity -> |phi| < 1
Q75: Describe the Akaike and Schwarz Criteria
A: Akaike information criterion (AIC) = exp(2k/T)*sum(et^2)/T Schwarz information criterion (SIC) = T^(k/T)*sum(et^2)/T They use different penalty factors The best model - information criterion attains minimum More commonly used SIC
Q83: Differentiate between Box-Pierce Q-statistic and Ljung-Box Q-statistic
A: Box-Pierce - used to test if time series is white noise - hypothesis: autocorrelations are jointly equal zero. - useful when large samples Ljung-Box - same methodology as Box-Pierce, but used for small samples Both are Chi-squared tests
Q30: Describe the central limit theorem (3 main properties)
A: CLT - for simple random samples of size n from population with mean mju and finite variance sigma^2, sampling dist approaches a normal probability dist as sample gets laege Exteemely useful - hypothesis testing (usually required n >= 30) Key properties of CLT - 3: *If n sufficiently large, sampling dist approximately normal *Mean of population = mean of the dist of all possible sample means *Variance of dist sample is sigma^2/n (population variance / sample size)
Q76: Describe conditions for consistency when selecting a model
A: Consistency used to compare different selection criteria. 2 conditions required When true model or data generating process is one of defined regression models, probab of selecting true model approaches 1 as sample size increases When true model is not one of the defined reg models, probab of selecting best approximation model approaches 1 as sample size increases The most consistent selection criterion - Schwarz information criterion (SIC) Akaike info crit (AIC) also useful - it is efficient asymptotically (SIC is not) - chooses reg model with one-step-ahead forecast error variances closest to variance of true model
Q24: Describe key properties of the continuous uniform distribution (3)
A: Cont Uniform dist defined over range between lower limit a and upper limit b. 3 major properties: *For all a<=x1 < x2<=b (for all x1 and x2 between boundaries a and b) *P(X < a or X > b) = 0 (probability outside boundaries equal zero) *P(x1 <= X <= x2) = (x2 - x1)/(b - a). It defines probability of outcomes between x1 and x2 f(x) - rectangle, F(x) - linear over variable's range F(x) = 1/(b-a) for a<= x <= b, else f(x) = 0 E(x) = (a+b)/2 Var(x) = (b-a)^2/12
Q19: What is covariance?
A: Cov(X,Y) - expected value of the product of the deviations of the two random variables. Hard to interpret
Q23: Describe and interpret the best linear unbiased estimator (BLUE) (4 properties)
A: Desirable properties of an estimator - 4: *Unbiased - E(X) of the estimator = parameter you try to estimate *Efficient - variance smaller than all other unbiased estimators *Consistent - accuracy of estimate increases as the sample size increases *Linear - estimator can be used as a linear function of sample data. Estimator is best, linear, unbiased -> BLUE
Q34: Describe a mixture distribution
A: Distribution can be combined to create unique probability density functions (useful if data doesn't follow 1 specific dist) 4 key properties: *Mixture distrib contain elements of both parametric and nonparametric dist. Input dist - parametric, weights of each dist are nonparametric *Skewness can be changed by mixing dists with different means *Kurtosis can be changed by mixing dists with different variances *Different levels of skew/kurtosis can reveal extreme events, that were previously difficult to identify
Q96: Describe the weighted moving average (EWMA) model in estimation of volatility
A: EWMA - specific case of genral weighting model. Main difference - weights assumed to decline exponentially back through time. sigma(n)^2 = lambda*sigma(n-1)^2 + (1-lambda)u(n-1)^2, lambda - weight on previous volatility estimate High values of lambda - minimize effect of daily percentage returns on volatility estimate Low value of lambda - increase the effect of daily percentage returns on volatility estimate Benefit of EWMA - requires few data points
Q97: Describe the generalized autoregressive conditional heteroskedasticity (GARCH(p,q)) model in estimation of volatility
A: GARCH(1,1) takes into account most recent estimates of variance, squared return and variable that accounts for a long-run average lvl of variance In GARCH(p,q): p - number of lagged terms on historical returns squared, q - number of lagged terms on historical volatility Formula for GARCH(1,1): sigma(n)^2 = w + alpha*u(n-1)^2 + beta*sigma(n-1)^2, alpha - weighting on previous period's return, beta - weighting on previous volatility estimate, w - weighted long-run variance = delta*V(L), V(L) - long-run average variance = w/(1-alpha-beta), alpha+beta+delta = 1, alpha+beta < 1 for stability, so that delta is not negative EWMA is a special case of GARCH(1,1) 0 w=0, alpha=1-lambda, beta = lambda
Q65: What is the effect of multicollinearity on regression analysis?
A: Greater probability that we will incorrectly conclude that a variable is not statistically significant (e.g. a Type II error)
Q70: Describe and interpret the joint hypothesis tests
A: H0: b1 = b2 = ... = bj = 0 | HA: one or more of bj !=0 F-statistic used (1-tail test): Fstat = ESS/k / SSR/(n-k-1) , ESS - explained sum of squares, SSR - sum of squared residuals
Q11: What is conditional probability
A: Here occurence of one event affects probability of occurence of another event P(A | B)
Q45: List critical z-values for one- and two-tailed test for lvls of signif: 10%, 5% and 1%:
A: I prepared a table to answer it: lvl of signif Two-tailed test One-tailed test 10% +/- 1.65 +/- 1.28 5% +/- 1.96 +/- 1.65 1% +/- 2.58 +/- 2.33
Q55: Describe the problem of heteroskedasticity (2 types)
A: If variance of residuals constant across observations - homoskedastic - good If opposite true - heteroskedasticity - bad. Two types: *Unconditional heteroskedasticity - not realted to level of independent variables - doesn't change value of X systematically - no major problems with regression *Conditional heteroskedasticity - related to level of independent variables. E.g. Exists if variance of residual increases as value of X increases. Significant problems for statistical inference
Q64: How to avoid dummy variable trap?
A: If we take n classes, we must use n-1 dummy variables (one of them will be held in intercept)
Q93: What is implied volatility?
A: Implied volatility of an option computed from option pricing model, such as Black-Scholes-Merton Volatility of asset not directly observed, so we compute implied volatility Index for publishing implied volatility - Chicago Board Options Exchange (CBOE) Volatility Index (VIX). Demonstrates volatility of 30-day calls and puts of S&P 500 Index. Often referred as the fear index - it reflects current market uncertainties
Q14: Compare independent and dependent events
A: Independent events are not affected by one another. P(A|B) = P(A) or P(B|A) = P(B) Dependent events if condition above not satisfied
Q39: Describe the hypothesis testing procedure (7 steps)
A: It consists of 7 steps: 1.State the hypothesis 2.Select appropriate test statistic 3.Specify lvl of signif 4.State decision rule regarding hypothesis 5.Collect sample and calculate sample stats 6.Make decision regarding hypothesis 7.Make decision based on result of the test
Q3: What is probability distribution?
A: It describes probabilities of all possible outcomes for a random variable. Sums to 1
Q79: What is white noise?
A: It is a condition, when time series is serially uncorrelated. Here mean = 0 and variance constant If observations in process independent - independent white noise If independent process also follows normal distribution - normal white noise/ Gaussian white noise Model's forecast errors should follow a white noise process
Q43: Describe confidence interval in hypothesis testing.
A: It is a range of values reasercher believes the true population parameter may lie. CI construction: (sample stat - (critical val * standard error)) <= population parameter <= (sample stat + (critical val * standard error))
Q9: Describe discrete uniform random variable
A: It is a variable for which probabilities for all possible outcomes are same (coin, fair die) F(xn) = n*p(x)
Q48: Describe the Chebyshev's Inequality
A: It states that for any set of observations, whether sample or popul data and regardless of shape of distribution The percentage of observations that lie within k sd of mean is at least 1 - 1/k^2 for all k > 1 Key relationships according to Chebyshev's Inequality for any distribution: 36% of obs lie within +/- 1.25 sd of mean 56% of obs lie within +/- 1.50 sd of mean 75% of obs lie within +/- 2 sd of mean 89% of obs lie within +/- 3 sd of mean 94% of obs lie within +/- 4 sd of mean
Q84: Describe the moving average process
A: MA process is a linear regression of current values of time series against both current and previous unobserved white noise error terms MA(1) process has mean 0, variance constant. Def -> y(t) = e(t) + theta*e(t-1), theta - coeff for lagged random shock If we want to express current observarion in terms of past observables - transformation needed. Condition |theta|<1, formula for inversion: e(t) = y(t) - theta*e(t-1) Inverten MA process - autoregressive representation (AR) Covariance stationarity essential to predictive ability of the model!
Q74: What is Mean Squared Error (MSE)?
A: MSE - statistical measure - sum of squared residuals divided by total number of observations MSE = sum(et^2)/T, T - total sample size MSE based on in-sample data. MSE closely related to R^2. Model with smallest MSE has the largest R^2 Selecting model based on min MSE bad for out-of-sample forecasts Unbiased estimate of MSE - s^2. It corrects MSE for degrees of freedom s^2 = sum(et^2)/(T-k) s^2 related to adjusted R^2
Q40: Compare and describe Null and Alternative Hypothesis
A: Null Hypothesis - H0 - hypothesis we want to reject. H0 indicates maintanence of status quo (contain =) Alternative Hypothesis - H1/HA - concluded if sufficient evidence to reject H0. Indicates significant change in status quo
Q33: Describe key properties of the F-distrubution (3)
A: Often used in hypothesis testing (e.g equality of the variances) F-test used under assumption that populations normally distributed and samples independent F = s1^2/s2^2 F-dist right-skewed, bounded below by zero, shape determined by 2 separate degrees of freedom 3 key properties of F-dist: *Approaches norm dist as number of observations increases *Random variable's t-value squared (t^2) with n-1 df is F-distributed with 1 df in numerator and n-1 df in denominator *Exists relationship between F- and chi-squared distributions: F = chi-squared / number of observations in numerator
Q32: Describe key properties of the Chi-Squared distrubution (3)
A: Often used in hypothesis testing (e.g population variance) 3 key properties: *Assymetrical *Bounded below by zero *Approaches norm dist in shape as df increase Test statistic = ((n-1)s^2)/sigma0^2 s^2 - sample variance, sigma0^2 - hypothesized value for the population variance
Q15: Describe the addition rule for probabilities
A: P(A or B) = P(A) + P(B) - P(AB) If events mutually exclusive - P(A or B) = P(A) + P(B)
Q35: Describe Bayes' Theorem
A: P(A|B) = (P(B|A) * P(A))/P(B) => P(A|B) = P(AB)/P(B) Bayes' theorem provides a framework for determining probability of one random event occurring given that another random event already occurred - conditional probability If two events highly correlated, conditional probability of the event occurring (e.g. bond A defaults given that bond B is in default) is always higher than unconditional probability of the e vent occurring
Q44: What is p-value?
A: P-value - probability of getting test stat that would lead to rejection of the null hypothesis, assuming null hypothesis true. Smallest lvl of signif for which null hypothesis can be rejected.
Q37: What is a posterior probability?
A: Probability set after seeing occurence of specific events (updated prior belief)
Q12: What is joint probability of two events and how to calculate?
A: Probability that they both occur P(AB) = P(A | B) * P(B)
Q61: How to calculate adjusted R^2?
A: Ra^2 = 1 - [(n-1)/(n-k-1) * (1-R^2)] If way too many Xi -> Ra^2 may be even negative
Q27: Describe key properties of the poisson distrubution
A: Random variable X refers to number of successes per unit Parameter lambda refers to the average or expected number of successes per unit P(X=x) = (lambda^x * e^-lambda)/x! E(X) = Var(X) = lambda
Q60: How to calculate standard error of the regression?
A: SER = sqrt(s(e)^2 = sqrt(SSR/(n-k-1) = sqrt(sum(ei^2)/(n-k-1)) The smaller SER, the better the fit
Q77: Describe sources of seasonality (5)
A: Seasonality is a pattern that tends to repeat from year to year - 5 main sources: *Specific examples - increases/decreases that occur at only certain times of the year *Weather *Economic activity (expand in 4th quarter, contract (kurczy się) in 1st) *Stochastic seasonality - annual approximate changes *Deterministic seasonality - annual exact changes
Q20: What is correlation?
A: Standardized measure to interpret the relationship between two variables Ranges from -1 to 1
Q66: How to detect multicollinearity?
A: The most common whey - none of coeffs significant, while R^2 high Also high correlation among indep vars (if > 0.7 - multicollinearity is a potential problem (if 2 variables)
Q16: What are measures of central tendency?
A: They identify the center, or average of a data set
Q78: What is and what are the conditions for a series to be covariance stationary (3 conditions)
A: Time series is covariance stationary when remains stable among its present and past values 3 conditions to be stationary: *Mean stable over time *Variance finite and stable over time *Covariance structure stable over time
Q25: Describe key properties of the bernoulli distribution
A: Two possible outcomes - success (1) or failure (0) p - prob of success p-1 - prob of failure Commonly used to assess if company defaults during specified time period
Q46: Describe the Chi-squared test
A: Used for hypothesis tests regarding variance of normally distrib population Sigma^2 - true pop var / sigma^2(0) - hypothesized var: H0: sigma^2 = sigma^2(0) vs HA: sigma^2 != sigma^2(0) (also combos with <=/ >=) Chi-squared test statistic = (n - 1)*s^2 / sigma^2(0)
Q8: What is inverse cumulative distribution function?
A: Used to find value that is connected to a specific probability E.g. we want to know specific x, where 15,67% of the distribution is less than or equal to x
Q71: Describe what is test of a single restriction
A: We use it if we want to test if one variable is equalt to another H0: b1 = b2, HA: b1 != b2
Q18: What is ezpected value?
A: Weighted average of the possible outcomes of a random variable Statistically, it is our "best guess" of the outcome of a random variable
Q81: Describe the Wold's theorem
A: Wold's theorem proposes a way to model process of removing trend and seasonal components from time series and isolating underlying covariance stationary process Form of the model => e(t) + b1*e(t-1) + b2*e(t-2) +... = sum(bi*e(t-i)) It is a general linear process e terms refer to innovations - errors that would result from a good forecast of covariance stationary process
Q50: Interpret a sample regression function
A: Yi = b0 + b1*Xi + ei
Q49: Demonstrate the process of backtesting
A: backtesting involves comparing expected outcomes vs actual data E.g. at 95% conf inter we expect that event exceeds it 5% of time Backtest common for VaR *VaR models fail to quickly react to changes in risk levels
Q82: Which bands used for autocorrelation functions to determine looking at graph if time series is white noise?
A: bands are +/- 2*sqrt(T) We would expect that 95% of sample autocorrelations and partial autocorrelations fall within this interval if time series is a white noise
Q31: Describe key properties of the student's t distrubution (4)
A: bell-shaped probab dist, symmetrical about mean Appropriate when small samples n<30, unknown variance and normal/approx normal distrib Key properties of Student's t dist - 4: *Symmetrical *Defined by a single parameter - degrees of freedom (df) = n-1 *More probability in tails (fatter tails) than in norm dist *As df larger, shape of t-dist approaches standard norm dist
Q69: How to build a Confidence Interval for a reg coef?
A: bj +/- (tc*s(bj)) = estimated reg coeff +/- (critical t-value)*(coeff stand error)
Q7: What is cumulative distribution function?
A: cdf - denoted F(x), defines probability that rand var X, takes on value less or equal to x Represents sum, cumulative value F(x) = P(X <= x)
Q28: Describe key properties (5) of the normal distrubution
A: central role in portfolio theory Probability density function - hard to write here Key properties - 5: *Norm dist completely described by mean and variance X~N(mju, sigma^2) *Skewness = 0 - symmetric *Kurtosis = 3 *Combo of normally distributed random variables is also normally distributed *Probabilities of outcomes above and below mean get smaller and smaller, but never equal zero 68% of outcomes within one standard deviation Approx 95% of outcomes within two sd 90% - 1.65*sd 95% - 1.96*sd 99% - 2.58*sd Standard normal distribution (z-dist) - standardized - mean 0 and sd 1 -> z= (observation - population mean) / standard deviation = (x - mean)/sd
Q53: Describe what is and how to calculate the coefficient of determination (+ concept of sum of squares)
A: coefficient of determination (R^2) is a measure of goddness of fit. It is a % of variation in Y explained by X Total sum of squares = explained sum of squares + sum of squared residuals Sum(Yi(obs) - Y(mean)^2 = Sum(Y(reg) - Y(mean)^2 + Sum(Yi(obs) - Y(reg)^2 TSS = ESS + SSR R^2 = ESS/TSS = Sum(Y(reg) - Y(mean)^2 / Sum(Yi(obs) - Y(mean)^2 R^2 = 1 - SSR/TSS = 1 - [Sum(Yi(obs) - Y(reg)^2 / Sum(Yi(obs) - Y(mean)^2] r - correlation coefficient between Xi and Yi -> r = sqrt(R^2)
Q99: What is correlation
A: correlation is a standardized measure of linear relationships between 2 variables p = <-1;1> p(X,Y) = cov(X,Y)/(sigma(X)*sigma(Y)) Cov(X,Y) = E[(X - E(X)) * (Y - E(Y))] = E(X,Y) - E(X)*E(Y)
Q22: What are coskewness and cokurtosis?
A: coskewness - 3rd cross central moment Cokurtosis - 4th cross central moment Many risk models overlook coskewness and cokurtosis. GARCH captures essence of themy (time-varying volatility/ correlation)
Q98: Explain the mean reversion process and how it is connected to GARCH and EWMA models
A: empirical data shows - volatility is mean-reverting GARCH tends to display better theoretical justification than EWMA However, outcomes are often inconsistent with model's assumptions (specifically, alpha+beta sometimes greater than one - instability in volatility estimation. Then analysts use EWMA models) Alpha+beta - persistence. If model is to be stationary the sum must be less than 1. persistence describes how quickly volatility will revert to its long-term value. The higher persistence, the longer it will take to revert to the mean. Persistene = 1 means there is no reversion, each change in volatility, a new lvl is attained *GARCH models estimated using maximum likelihood techniques
Q5: Describe and compare discrete and continuous distribution
A: for discrete distribution p(x) = 0 if x cannot occur or p(x) > 0 if x can occur For continuous distribution p(x) = 0 even though x can occur. Here we use ranges - P(X1 <= x <= X2)
Q54: Describe confidence intervals for regression coefficients
A: for regression coefficient B1: b1 +/- (t(c) * s(b1)) or [b1 - (t(c) * s(b1)) < B1 < b1 + (t(c) * s(b1))], t(c) - critical 2-tail t-value for selected confidence lvl, s(b1) - standard error of regression To test significance of B1: if 0 is not included in CI -> coefficient is significantly different from zero Also you can test if coefficient is equal to some hypothesized value: T = (b1 - B1)/s(b1) -> b1 - coef from regression, B1 - hypothesized value
Q52: Describe the assumptions of OLS (9)
A: there are 9 key properties/assumptions: *Expected value of the error term is 0 *All (X,Y) observations independent and identically distributed *Unlikely that large outliers will be observed in data (potential to create misleading results of regression) *Linear relationship exists between X and Y *Model is correctly specified (appropriate indep var and does not omit variables) *Independent variable uncorrelated with the error term *Variance of error term constant for all Xi *No serial correlation of error terms exist *Error term normally distributed
Q95: Explain how various weighting schemes can be used in estimating volatility
A: there are a couple of popular ways to construct weighting scheme Mean return of individual return: u= 1/m*sum(u(n-1)), m - number of observations If we assume mean return 0, maximum likelihood estimator for variance -> sigma(1|n) = 1/m*sum(u(n-1)^2) If we want weight recent data more heavily - sigma(n)^2 = sum(alpha(i)*u(n-1)^2), alpha(i) - weight on return i days ago. Alphas must sum to 1 Most frequently used method with more weight to recent data -> autoregtressive condiitonal heteroskedasticity model (ARCH(m)) -> sigma(n)^2 = delta*V(L) + sum(alpha(i)*u(n-1)^2) with delta + sum(alpha(i)) = 1, so that sigma(n)^2 = w + sum(alpha(i)*u(n-1)^2), w = delta*V(L) - long-run variance weighted by parameter delta
Q89: Define and describe the properties of autoregressive moving average (ARMA) process
A: time series can show signs of both processes - MA and AR - e.g. stock prices migh show evidence of being influenced by unobserved shocks (MA component) and their own lagged behaviour (AR component) ARMA process - formula: y(t) = phi*y(t-1) + e(t) + theta*e(t-1) ARMA to be covariance stationary, still condition |theta| < 1 ARMA can be extrapolated to ARMA(p,q) model -> p-AR portion, q-MA portion
Q42b: Describe type I and II errors
A: type I error - rejection of H0 when it is true type II error - failure to rejection of H0 when it is false Usually we wish to use the test statistic that is most powerful -> max(1 - beta)
Q47: Describe the F-test
A: used to hypotheses concerned with equality of variances of two populations. Follows F-distribution (assumption: populations normally distributed and samples independent) H0: sigma^2(1) = sigma^2(2) vs HA: sigma^2(1) != sigma^2(2) (also combos with <=/ >=) Test statistic - ratio of the sample variances: F= s^2(1) / s^2(2) -> always put larger variance in numerator - then you have to consider only critical value for right-hand tail
Q91: What is volatility?
A: volatility of variable - sigma - standard deviation of the variable's continuously compounded return Options - one-year period Risk management - one-day period Calculation of continuously compounded return -> u(i) = ln(S(i)/S(i-1)), S(i) - asset price at time i Calculation of proportional change in an asset -> u(i) = (S(i) - S(i-1))/S(i-1). From risk management perspective, daily volatility usually refers to sd of the daily proportional change in asset value If you want to expand time horizon - sigma*sqrt(T), T - amount of days to expand on
Q73: Describe OLS trend model to estimate and forecast trends
A: y(t) = beta0 + beta1(t)
Q85: Show how looks MA(q) process
A: y(t) = e(t) + theta1*e(t-1) + ... + thetaq*e(t-q) Covariance stationarity essential to predictive ability of the model!
Q88: Show how looks AR(p) process
A: y(t) = phi1*y(t-1) + phi2*y(t-2) + ... + phip*y(t-p) + e(t)
Q42a: Differentiate between a one-tailed and a two-tailed test
One-tailed - used when H0: u >= u0/ u <= u0 | HA: u < u0 / u > u0 Two-tailed - used when H0: u = u0 | HA: u != u0
Q13: How to calculate conditional probability of A given B?
P(A|B) = P(AB)/P(B)
Q36: Compare the Bayesian and frequentist approaches
A: frequentist approach: drawing conclusions from sample data based on the frequency of that data. Probability of a positive event will be 100% if the sample data contains only positive events. Simply based on observed frequency of positive events occurring. Questionable when small sample size Often used with larger sample sizes Easier to understand and implement than Bayesian Bayesian approach: Instead based on a prior belief regarding the probability of an event occuring. Updates the probabilities on the new information Weakness in relying on prior beliefs. The prior assumptions often based on frequentist approach/ subjective analysis Often used with small sample sizes
Q29: Describe key properties of the lognormal distrubution (2)
A: generated by the fun exp(x) f(x) - hard to write 2 key properties: *Lognormal dist skewed to the left *Bounded from below by zero - useful for assets prices (always positive values)
Q10: What is uncdonditional probability
A: i.e marginal probability. Refers to probability of an event regardless of past or future occurrence of other events
Q80: Describe the lag operator
A: it is L -> y(t-1) = L*y(t) Same y(t-2) = L*y(t-1) = L^2*y(t) L^2*y(t) = y(t-m)
Q51: Define an ordinary least squares (OLS) regression
A: it is a process that estimates population parameters Bi that minimize the squared residuals: Minimize sum(ei^2) = sum(Yi - (b0 + b1*Xi)^2) B1 = Cov(X,Y)/Var(X) B0 = Y(mean) - b1*X(mean)
Q94: Describe the power law
A: it states, that when X is large, value of variable V has following property: P(V>X) = K*X^-alpha V - variable X - large value of V K and alpha - constants By taking ln - we can perform regression analysis to determine power law constants, K and alph: Ln[P(V>X)] = ln(K) - alpha*ln(X) This law suggests that extreme movements have very low probability of occurring, but still higher than what is indicated by normal distribution
Q21: Describe mean, variance, skewness and kurtosis in relation to central moments
A: mean - first raw moment First central moment = 0 Variance - second central moment Skewness - standardized third central moment -> 3rd central moment/ sigma^3. Refers to extend to which distribution is not symmetric around mean. Norm dist = 0 Kurtosis - standardized fourth central moment -> 4th central moment/ sigma^4. Refers to how fat or thin are the tails in data dist. Norm dist = 3
Q67: How to correct multicollinearity?
A: most common - omit one or more correlated variables (possible to use stepwise regression)
Q63: Explain the concept of multicollinearity (2 types):
A: multicollinearity - condition when two or more indep vars are highly correlated with each other (or combinations of indep vars) It distorts standard error of regression and coeff se, - problems with t-tests for statistical significance Taking degree of correlation - 2 types of multicollinearity *Perfect multicollinearity - one indep var is a perfect linear combination of other indep vars. Impossible to find OLS estimators for regression *Imperfect multicollinearity - 2 or more indep vars highly correlated, but less than perfect. This requires detecting and correcting
Q1: Describe discrete random variable
A: number of possible outcomes can be counted, For each outcome there is measurable and positive probability
Q2: Describe continuous random variable
A: number of possible outcomes is infinite, even when lower/upper bounds exist
Q26: Describe key properties of the binomial distribution
A: number of successess in a given number of trials p - prob of success, constant for each trial and trials independent Final outcome - number of successes in a series of n trials p(x) = P(X=x) = (number of ways to choose x from n)*p^x(1-p)^(n-x) numbers of ways to choose x fromm n = n!/((n-x)!x!) So, p(x) = n!/((n-x)!x!)*p^x(1-p)^(n-x) E(X) = np Var(X) = npq, q = 1-p Binomial distribution often used in the process of asset valuation - success/failure in investment
Q17: When do we use geometric mean?
A: often used when calculating investment returns over multiple periods, or measuring compound growth rates
Q4: What is probability function?
A: p(x) - specifies the probability that a random variable is equal to a value Used for discrete distribution p(x) = P(X=x) 0<=p(x)<=1 Sum(p(x)) = 1
Q6: What is probability density function?
A: pdf - function denoted f(x), used to generate probability of outcomes of continuous distribution Equivalent of a probability function for cont dist
Q59: What is omitted variable bias (2 conditions)?
A: present when two conditions met: *Ommited variable correlated with the movement of independent variable in model *Omitted variable is a determinant of the dependent variable Bad situation - error terms will be correlated with Y (because of OVB) To handle this we use multiple regression
Q87: How to calculate the t-period autocorrelation?
A: rho(t) - phi^t for t = 0,1,2,...
Q57: Describe how to correct the heteroskedasticity
A: robust standard errors - on exam if you see the heteroskedasticity is a case - use robust se
Q92: What is variance rate?
A: square of volatility. - it increaes in linear fashion over time -> sigma^2*T
Q68: How to calculate t statistic for test of significance?
A: t = (bj - Bj)/s(bj) = (estimated reg coeff - hypothesized value) / coeff stand error
Q41: Describe how to calculate the test statistic
A: test statistic = (sample statistic - hypothesized value)/standard error of the sample statistic
Q62: Explain assumptions of multiple linear regression model (6)
A: there are 6 key assumptions: *Linear relationship exist between dependent and independent variables *Indep var not random - no exact linear relation between any two or more indep var *Expected value of error term is 0 *Variance of error term constant for all observations *Error term for 1 observ not correlated with error term for 2. *Error term normally distributed
