Bionic Turtle

[Learn] Transition matrix [practice, credit]

2009-07-09T23:20:24+00:00

Question [source 2009 sample Full 1]: E1.41. Consider the following one‐period transition matrix:

[source] 41a. If a company is originally in State A, what is the probability that the company will have defaulted strictly before the fourth transition period from now?
[my adds] 41b. De Servigny writes that “default is an absorbing state.” How does this manifest in the transition matrix?
41c. Which of the credit risk portfolio models (reviewed in the FRM) rely on a credit transition/migration matrix as an input?
41d. Assume the answer to 41a (above) represents the three-year cumulative probability of default (PD). If we now ignore the transition matrix and assume each year’s marginal probability of default (marginal PD) is constant, what is the implied marginal (annual) PD?
41e. Assume the riskless yield is 4%. Based on the marginal PD, what is the implied spread under the assumption of 0% recovery? what is implied spread under assumption of 70% recovery (30% LGD)?

Answer here in forum or here in wiki.

[Learn] Gujarati 07.08 [practice, quant]

2009-07-09T17:38:42+00:00

07.08 The characteristic line of modern investment analysis involves running the following regression:

where

r = the rate of return on a stock or security
rm = the rate of return on the market portfolio represented by a broad market index such as S&P 500, and
t = time.

In investment analysis, B2 is known as the beta coefficient of the security and is used as a measure of market risk, that is, how developments in the market affect the fortunes of a given company.

Based on 240 monthly rates of return for the period 1956 to 1976, Fogler and Ganapathy obtained the following results for IBM stock. The market index used by the authors is the market portfolio index developed at the University of Chicago:

a. Interpret the estimated intercept and slope
b. How would you interpret r^2?
c. A security whose beta coefficient is greater than 1 is called a volatile or aggressive security. Set up the appropriate null and alternative hypotheses and test them using the t test. Note: Use α = 5%.

Answer here in forum or here in wiki.

[Learn] 1st Focus Review (topics 1 & 2) on Saturday: Agenda

2009-07-08T19:13:27+00:00

For customers attending Saturday’s webinar (details are here under announcements), I have attached a draft of the presentation I will be using (please note: I won’t use all of the slides, some are just backup in anticipation of some questions). If you have time, please direct your attention to the five questions/exercises.

The Exercises are introduced with a red banner (“Exercises”) and they have the following icon in the upper right-hand corner:

Draft presentation

Here is link to PDF

I have carefully analyzed the 2009 FRM AIMs and the sample exams. Based on this, and limited time, I have reduced to the following “focus themes:”

Value at risk (CDF quantile)
CAPM/RAPM
Sample estimators (inference)
Regression
MCS
Volatility
Select distributions

Warning: this is a lot to cover in a relatively short period of time. Our goal for the webinar will be “merely practical:” we are anticipating the exam and, in my opinion, the key things you need to know for the exam (in regard to these first two topics, anyway!). Therefore, questions should be limited to our scope because we will be going pretty fast!

The best thing you can do is look at (review) the five questions/exercises. I have carefully chosen/developed these questions for maximum impact (most are expansions of questions in the sample exam):

Portfolio VaR
Fund RAPMs
Fund performance
GE vs. S&P 500 (regression)
EWMA & GARCH(1,1)

If you cannot attend, don’t worry: we will save recording and the edited (final) presentation to the member page…Suzanne and I look forward to “seeing” you Saturday. Thanks!

[Learn] DV01 vs Maturity [practice, market]

2009-07-08T16:32:44+00:00

[Source 2009 FRM Full Exam 1] 37. Assuming other things constant, bonds of equal maturity will still have different DV01 per USD 100 face value. Their DV01 per USD 100 face value will be in the following sequence of highest value to lowest value:

a. Zero coupon bonds, par bonds, premium bonds
b. Premium bonds, par bonds, zero coupon bonds
c. Premium bonds, zero coupon bonds, par bonds
d. Zero coupon bonds, premium bonds, par bonds

[my adds]

37b. What is the difference between DV01 and modified duration?
37c. What is the impact of maturity on DV01 (e.g., increasing)?
37d. What is the impact of yield (YTM) on DV01

My answers here in forum or here in wiki.

[Learn] Gujarati 07.07 [practice, quant]

2009-07-08T14:51:47+00:00

Question 07.07: Based on the data for the years 1962 to 1977 for the United States, Dale Bails and Larry Peppers obtained the following demand function for automobiles:

Ŷ(t) = 5807 + 3.24Xt, R^2 = 0.22
standard error (intercept) = 1.634

where:

Y = retail sales of passenger cars (thousands) and
X = the real disposable income (billions of 1972 dollars).
Note: The standard error (se) for b1 is not given.

a. Establish a 95% confidence interval for B2 (the intercept).
b. Test the hypothesis that this interval includes B2 = 0. If not, would you accept the null hypothesis?
c. Compute the t value under H0:B2 = 0. Is it statistically significant at the 5 percent level? Which t test do you use, one tailed or two-tailed, and why?

Answers:

[Learn] Gold lease rate [practice, market]

2009-07-07T18:39:32+00:00

[source 2009 FRM sample Full 1] E1.19. If the gold lease rate is higher than the risk‐free rate, what is the market structure of the forward market for gold?

a. Contango
b. Backwardation
c. Inversion
d. Need more information to determine

[next are my adds, let’s beat up Hull and McDonald!]

Assume the following:

The risk-free rate is 4%
The spot price of gold is $935 USD/ounce and
The six month (Dec 2009) forward contract price is $930 USD/ounce (approximately true).
Further, assume the size of one gold futures contract is 100 ounces.

E1.19b Assume the correlation between spot and future price change is (conveniently) perfect at 1.0. And, the volatility of (change in) gold spot price is 20% but the futures is more volatile at 40%. If a gold producer plans to sell one million ounces in the future, what is the optimal hedge?

E1.19c. If gold has no correlation with the (stock) market, what does CAPM imply for the expected growth rate of gold?

E1.19d. What is gold’s implied lease rate (continuous compound frequency)?

E1.19e. [hard] What is the implied commodity discount rate (continuous)?

E1.19f. What does McDonald say about the “typical” gold futures curve; i.e., characterized by…?

E1.19g. Which characterizes the situation?

contango and normal contango
contango and normal backwardation
backwardation and normal backwardation
backwardation and normal contango

Answers here in forum or here in wiki.

[Learn] Gujarati 07.02 [practice, quant]

2009-07-07T16:50:09+00:00

Question 07.02: State with brief reasons whether the following statements are true, false, or uncertain.

a. OLS in an estimating procedure that minimizes the sum of the errors squared (as below)
b. The assumptions made by the classical linear regression model (CLRM) are not necessary to compute OLS estimators.
c. The theoretical justification for OLS is provided by the Gauss-Markov theorem
d. In the two-variable PRF, b2 is likely to be a more accurate estimate of B2 if the disturbances ui follow the normal distribution.
e. The OLS estimators b1 and b2 each follow the normal distribution only if ui follows the normal distribution.
f. R^2 is the ratio of TSS / ESS.
g. For a given alpha and d.f., if the computed ItI exceeds the critical t value, we should accept the null hypothesis.
h. The coefficient of correlation, r, has the same sign as the slope coefficient b2.
i. The p value and the level of significance, α, mean the same thing.

Answers here in forum or here in wiki.

[Tools] 2009 4.a.ii Valuation and risk models (VaR) - sample

2009-07-06T23:05:55+00:00

Linda Allen Chapter 3 (Putting VaR to work) and Chapter 5 (Extending VaR to operational risk)

*[Tools] 2009 4.a.ii Valuation and risk models (VaR)

2009-07-06T22:59:40+00:00

[Learn] Transfer credit risk [practice, credit]

2009-07-06T18:12:54+00:00

This sample question is easier than will be the actual exam. As usual, I added some follow up questions to give you a Monday stretch - David

[source Full Sample I. E01.10] A bank is considering ways of significantly reducing or eliminating its credit exposure to defaults on a loan portfolio so that the bank’s shareholders do not absorb the losses arising from such defaults. Ignoring institutional issues (e.g., tax, accounting, capital requirements), three of the following programs have a similar impact on the credit risk of the bank. Which alternative fails to reduce credit risk?

a. Sell the loan portfolio in its entirety to another bank.
b. Borrow to finance an additional risk reserve to supplement existing loan‐loss reserves.
c. Securitize the loan portfolio.
d. Buy credit protection on the loan portfolio with credit default swaps.

My adds:

The sample answer says “All three of the other [incorrect] choices are economically equivalent.” Is this true?
Compare the total risk(s) transferred by outright loan sale (a) to the use of credit default swaps (d).
The question refers to the motivation to transfer credit risk. What else can motivate securitization of credit-sensitive assets?
If the bank succeeds in a “true sale” of the credit-sensitive assets in a securitization, who will the assets be sold to? What are the accounting and legal benefits of this true sale?
If the bank decides to use the loans to originate a collateralized debt obligation (CDO), what is the essential difference between a cash CDO and synthetic CDO?

My answer here in forum or here in wiki.

[Learn] Gujarati 06.21, OLS regression [practice, quant]

2009-07-06T17:06:59+00:00

Question:
06.21 Table 6-15 gives data on verbal and math S.A.T. scores for both males and females for the period 1967-1990.
http://public.sheet.zoho.com/public/btzoho/table6-15
a. You want to predict the male math score (Y) on the basis of male verbal score (X). Develop a suitable linear regression model and estimate its parameters.
b. Interpret your regression results.
c. Reverse the roles of Y and X and regress the verbal score on the math score. Interpret this regression.
d. Let a2 be the slope coefficient in the regression of math score on the verbal score score and let b2 be the slope coefficient of the verbal score on the math score. Multiply these two values. Compare the resulting value with the r² obtained from the regression of math score on verbal score or the r² value obtained from the regression of verbal score on math score. What conclusion can you draw from this exercise?

[my adds]

I used Excel’s regression add-in to produce the ANOVA table below; I regressed Male Math scores (explained) on Male Verbal scores (explanatory). The OLS regression equation produced is: MATHM = 262.8 + 0.5386 * VERBM. Here is the ANOVA table produced by Excel:

e. How many estimators are there in this two-variable (univariate) regression?
f. What distribution characterizes all three estimators, including d.f., and why?
g. Show how the 95% confidence interval for each coefficient (slope and intercept) is calculated.
h. Test your hypothesis that the true slope (i.e., of the PRF function) is 0.5.
i. What is the standard error of regression (SER) and what does it mean?
j. The coefficient of determination is given as 0.8842. Confirm this by using the ANOVA table to calculate the R^2.
k. What does the F value signify?

Answers here in forum or here in wiki.

[Tools] 2009 4.a.i Valuation and risk models (VaR) - sample

2009-07-06T01:13:54+00:00

Linda Allen Chapters 2, 3 and 5. Introduction to value at risk (VaR).

[Learn] Return aggregation & VaR [valuation]

2009-07-06T02:02:11+00:00

I was tired of just talking about Linda Allen’s return aggregation & VaR (Chapter, p 61) so I just created a new learning spreadsheet (4.a.6 L Allen’s Return Aggregation) to illustrate (for customers: I will review in 4.a.ii tutorial; and this will go in the study notes, of course). I think it works to illustrate her “third way” to aggregate VaR. I tagged it yellow; i.e., not essential for exam but maybe helpful.

The inputs are really simple, just to illustrate: a portfolio with four assets (A, B, C, D), a historical window of only 5 days (daily returns) and the *current* portfolio weights:

1. Historical Simulation

Apply current portfolio weights to each historical day (each day is a column vector of returns); in this case, that simulates five days of portfolio returns. Sort and lookup, Excel = PERCNTILE(), to find VaR.

2. Variance-Covariance

Extract covariance matrix out of historical asset returns…again, here I showed how you get from correlations to covariance matrix to portfolio variance….

3. “Third Alternative”

Linda’s Allen’s third approach (see Fig 2.12) is to impose normal distribution on the aggregated returns. In his case, we have a series of 5 days of simulate returns (above; i.e., what if we held the current weights over the historical period?) and we grab the mean and variance from that series. Note I solved for the absolute not relative VaR; absolute VaR is consistent with PERCENTIILE() function,

*[Tools] 2009 4.a.i Valuation and risk models (VaR)

2009-07-05T21:18:59+00:00

Linda Allen Chapters 2, 3 and 5. Introduction to value at risk (VaR).

[Learn] 4.a.i. Tutorial, covariance matrix, relative/absolute value at risk (VaR)

2009-07-05T21:44:01+00:00

I started uploading 4.a.i tutorial (valuation and risk models). This tutorial matters because it is the FRM’s introduction to value at risk (VaR). Jack, who has a penchant for spotting key issues, recently asked two great questions about relative/absolute VaR and the covariance matrix. This made us realize the curriculum forgets to introduce both ideas; as I’ve suggested before, I still recommend Jorion’s Chapter 5 to “fill some gaps” although it’s not on the reading list.

So I added two learning spreadsheets:

4.a.1 Two-asset VaR, relative vs. absolute
4.a.2 Delta normal VaR

4.a.1 Two-asset VaR, relative vs. absolute

This is a compact sheet. I hope you analyze it, because I packed a lot of “stuff you should know” into one sheet! Given two assets, we compute four VaRs (in green). You should know:

How to compute two asset volatility
How to scale the volatility into VaR; i.e., multiply by deviate (1.645 @ 95% normal, 2.33 @ 99% normal)
How to scale the volatility/VaR from one day to n-days per the square root rule
That the scaling by square root requires i.i.d (not as sometimes said, normality. Scaling here follows from a stable distribution of i.i.d. returns. The normal is not the only stable function. See Rachev. Can you recall the other two?)

Then I added relative VaR and absolute VaR:

Relative VaR is “relative to the expected final wealth” so it does not use the expected (mean ) return. Jorion favors relative VaR because it is “conservative;” i.e., it will always be higher than absolute VaR given positive expected return.
Absolute VaR is “relative to the initial portfolio” or the baseline of zero, if you like. Put another way, expected gains offset worst expected losses.

I also added same set of VaRs to incorporate return autocorrelation. Positive autocorrelation increases the scaled volatility and therefore, the scaled VaR. You can connect this to the (always giving doubts) mean reversion in Linda Allen. Two sorts of mean reversion:

Mean reversion in volatility: just like the mean reversion in GARCH(1,1). The volatility is pulled toward a long-run average variance.
Mean reversion in returns: this is the violation of return i.i.d. that we require for the square root rule. This is “not independent” returns. Or, this mean reversion (in returns) is negative auto-correlation or negative serial correlation! In the example XLS above, my input is a positive autocorrelation, so the AR(1) VaR is higher. But if you switch the input to negative (e.g., –0.3 autocorrelation of returns), you will be simulating Linda Allen’s second type of mean reversion (in returns). And the above VaR will be smaller than the i.i.d. VaR. Consistent with her statement that “mean reversion in returns overstates true long run volatility.”

4.a.2 Delta normal VaR

This learning spreadsheet is my (color coded) re-creation of Jorion’s delta normal VaR in the FRM Handbook, Table 15.7 (exact same as prior handbook, like most of the handbook contents…). I tagged this red because it’s more advanced. In a nutshell, he maps a currency forward to three risk factors (spot exchange rate, domestic interest rate, and foreign interest rate) and then uses a 3x3 covariance matrix to calculate portfolio volatility. I will review this, in detail, in an upcoming “focus bag” video but wanted to share three things:

First, the second sheet has a very simple covariance matrix. I have a 7 minute video here on youtube that walks through this sheet. The inputs are three asset volatilities and three correlations. The point is, the covariance matrix is the product of [diagonal matrix of volatilities]*[correlation matrix]*[same diagonal of volatilities]. As such, this matrix covariance is simply the matrix equivalent of our familiar two-asset COVARIANCE = CORRELATION*VOLATILITY*VOLATILITY:

Two asset covariance (A,B) = correlation(A,B)*Volatility(A)*Volatility(B)
n-asset covariance matrix (Sigma) = diagonal matrix of volatilities * correlation matrix * diagonal matrix of volatilities

Second, we can move from two-asset to n-asset VaR with matrices. The spreadsheet (4.a.2) maps a currency forward to three risk factors, and the positions in the factors are given by vector (x). Given covariance matrix (Sigma), the portfolio variance and portfolio VaR are given by:

Third, in the XLS, on the right sidebar, I show the same derivation of the covariance matrix given the correlation matrix as an input.

In summary,

If (R) is the correlation matrix and (D) is the diagonal matrix that contains the volatilities, the covariance matrix (C) = (D)(R)(D)
Then given column vector (x) of portfolio positions, the portfolio variance = (x’)(C)(x)
So, we can express portfolio variance as a function of volatilities (D), correlation matrix(R) and positions (x): Portfolio Variance = (x’)(D)(R)(D)(x)

[Learn] VaR in a picture & explained in six words

2009-07-04T00:08:10+00:00

Like I did with a volatility chart, I updated our value at risk (VaR) chart for the video tutorials. If you want a clean breakdown of VaR approaches, this brief primer by Suna Reyent is very good, with interesting comments. Here is the chart I am using in the valuation and risk models tutorials (topic 4, 2009 FRM):

Here are my goals in summarizing this way:

At top, to remind the user has two “design decisions:” significance (1-confidence) and time horizon
Following Linda Allen (FRM assigned), three VaR approaches: parametric, non-parametric and hybrid. Parametric VaR is synonymous with Analytical VaR, and refers to the use of a probability distribution to characterize returns; e.g., normal, Levy for heavier tails. Some authors call this approach delta-normal but that’s imprecise because we do not require a normal assumption under parametric VaR (yet has an incredibly long life as a straw man). Rather, delta-normal is but one type of parametric VaR.
Within non-parametric, I am following Dowd (FRM assigned) with one category that includes all the simulations. So, you can see how it’s fine to say that VaR can broadly be approached two ways: parametric (with distributions) or simulated. My published stuff follows others in saying there are three approaches: parametric, historical simulation (simulation backwards) and Monte Carlo simulation (simulation forward). In this way, it’s semantic as to whether there are two or three approaches: in fact, there are several dozen approaches. Simulations are a huge topic, but I tag here just the three (3) that interest the FRM candidate: historical sim, bootstrap (a variation that might best be thought of as “with replacement,” as in, historical returns are replaced back into the historical pool, to be used again in the simulation), and Monte Carlo simulation (a field unto itself which is not mutually exclusive of other methods, but shown this way merely for convenience)
The hybrid (a.k.a., semi-parametric) features Linda Allen’s HS+EWMA (members can see this “in action” as a learning spreadsheet here.)
I included EVT (again, following Dowd) but have it shown in a different color because you wouldn’t normally think of EVT as a type of VaR approach; but in our limited study (GPD, GEV) the EVT distributions characterize the extreme tail and inform VaR quantile (i.e., what is the 99.9% “extreme” VaR) so I think it’s okay.

VaR in six words

My favorite all-time explanation of VaR was written by our assigned Kevin Dowd. Here it is: “A VaR is merely a quantile.” (p 32, Market Risk, 2nd Ed). Quantile has a specific meaning of course. And, in regard to “how do we arrive at VaR?” and “what do we do with it?” those are not succinct. I realize it’s not enough to know for the exam…but it maybe puts into perspective the voguish critique of VaR; it’s like criticizing the Excel PERCENTILE() function (actually, that is Excel’s built-in VaR function!), or ROE, or RAROC for that matter. The metric itself can’t be dangerous, can it?

[Learn] Implied LGD from bond rates [practice, credit]

2009-07-03T20:39:45+00:00

This is a good sample question from GARP: first, it is not too easy and not too hard; this IMO is about typical of the exam's difficulty and (ii) it does not favor formula memorization: the cited reference is de Servigny, but I do not think you will find the formula in the referenced reading. A formula is not needed, we can apply logic and yet another instance of a no-arbitrage idea - David

9. Suppose the rate on Company A’s one-year zero-coupon bond is 10.0% and the one-year T-bill rate is 8.0%. Assume the T-bill is riskless and the probability of default of Company A’s bond is 10%. What is the LGD of Company A’s bond?

a. 18.18%
b. 81.82%
c. 20.01%
d. 79.99%

[my adds next, let’s beat this up!]

9b. [hard] Assume instead we are given that the LGD of Company's A's bond is 75% (using Basel II LGD for junior debt under foundation IRB approach). Assuming the credit spread remains 2% (10% - 8%), what is the implied probability of default (PD)?

9c. The answer assumes annual compounding. If we instead assume continuous compounding (i.e., risky bond returns 10% continuous and T-bill returns 8% continuous), what is the implied loss given default (LGD)?. Notice that compounding matters; semi-annual compounding would give a slightly different result.

9d. [hard] As Saunders writes, "Collateral requirements are a method of controlling default risk; they act as a direct substitute for risk premiums in setting required loan rates." Let collateral (recovery) = 1 - Loss given default (LGD) and solve for the credit spread as a function of the probability of default (PD) and the recovery (c).

9e. Which continuous probability distribution--that is reviewed in the assigned Rachev (Fat-tailed and Skewed Asset Return Distributions)--are we most likely to find in use to characterize the loss given default (LGD) or recovery function?

9f. How are recovery rates (1-LGD) estimated?

9g. What are the most important determinants of recovery/LGD?

My answers here in forum or here in wiki.

[Learn] Gujarati 06.04 & 06.05 [practice, quant]

2009-07-03T19:21:09+00:00

Question 06.04: State whether the following statements are true, false, or uncertain. Give your reasons. Be precise.
a. The stochastic error term ui and the residual term ei mean the same thing.
b. The PRF gives the value of the dependent variable corresponding to each value of the independent variable.
c. A linear regression model means a model linear in the variables.
d. In the linear regression model the explanatory variable is the cause and the dependent variable is the effect.
e. The conditional and unconditional mean of a random variable are the same thing.
f. In Eq. (6.2) the regression coefficients, the Bs, are random variables, whereas the bs in Eq. (6.4) are the parameters.
g. In Eq. (6.1) the slope coefficient B2 measures the slope of Y per unit change in X.
h. In practice, the twp-variable regression model is useless because the behavior of a dependent variable can never be explained by a single explanatory variable.
i. The sum of the deviation of a random variable from its mean value is always equal to zero.

Question 06.05: What is the relationship between
a. B1 and b1
b. B2 and b2
c. ui and ei? Which of these entities can we observe and how?

Answers

Answer to 06.04 here in forum or here in wiki
Answer to 06.05 here in forum or here in wiki

[Learn] Merton model [practice, credit]

2009-07-02T21:30:06+00:00

The Merton model is a classic in credit risk. If you are interested in a few resources from last year: here is a 10-minute video I recorded last year to introduce the role of d2 in the Merton model for credit risk; and another with a bit more context (including how Merton is a so-called structural type of model) - David

[source FRM sample 2009 Full Exam 1] Question Full E1.08: You don’t have access to KMV’s data. Your boss wants you to tell him your estimate of the probability of default of a credit. To do so, you use the Merton Model because the credit you are considering has no systematic risk. In Merton’s Model, the distance to default (DD) and the expected default frequency (EDF) are:

a. positively and linearly related
b. negatively and linearly related
c. positively and non-linearly related
d. negatively and non-linearly related

my adds [some of these are tough]

E1.08e. Explain the Merton Model in a few brief sentences.
E1.08f. Cite two differences between Merton Model and Moody's KMV.
E1.08g. Cite a few variables that would decrease the EDF?
E1.08h. Cite a disadvantage of this approach (i.e., equity-based model of default prediction).
E1.08i. [hard] The question implies that the Merton Model requires, or wants, an assumption that the credit has no systematic risk. Is this true?
E1.08j. [hard] As the relationship between DD & EDF is non-linear, can we be more specific: what is the distribution of DD? Reconcile this distribution with the lognormal property of stock prices.
E1.08k. The answer says the risk-neutral probability of default (PD) = 1 – N(d2). But de Servigny says PD = N(-d2). Which is correct?
E1.08l. If the distance to default (DD) is 2.0, what is Merton's implied risk-neutral probability of default (PD)?
E1.08m. [hard] Under risk-neutral valuation, can we assume this PD applies in the real world?

My answers here in forum or here in wiki.

[Learn] Gujarati 06.01 & 06.02 & Greek dart boards [practice, quant]

2009-07-02T20:06:03+00:00

It's hard to study definitions, I think. But studying terms in Gujarati is helpful: his precision forces us to contend with ideas. The definitions below (first question in Chapter 6) constitute a great list of terms. My favorite single theme in Gujarati is estimator-versus-parameter (or sample:population). Many of the ideas here are detail on the single idea that many observed samples are drawn from the one unobserved population; that these samples must vary even as the population is static; and that therefore the sample estimators are themselves random variables (and really, most of the time, the central limit theorem tells us these estimators are asymptotically normal; not by accident do all three of the sampling distributions converge to the fourth sampling distribution, the normal) - David

Question 06.01: Explain carefully the meaning of each of the following terms:

a. Population regression function (PRF).
b. Sample regression function (SRF).
c. Stochastic PRF.
d. Linear regression model
e. Stochastic error term (ui).
f. Residual term (ei).
g. Conditional expectation.
h. Unconditional expectation.
i. Regression coefficients or parameters.
j. Estimators of regression coefficients.

[my adds]

k. What does “Stochastic” in “Stochastic PRF” mean, why does it matter?
l. Is linear regression model redundant, does not regression model imply “linear?”
m. What is the difference between the error and the residual?
n. Given the linear SRF: Y (est) = intercept + slope (X), where is the conditional expectation and the unconditional expectation?
o. In a two-variable regression, how many “estimators of regression coefficient” are there?

Question 06.02: What is the difference between a stochastic population regression function (PRF) and a stochastic sample regression function (SRF)?

Answers

To 06.01 here in forum or here in wiki (including, as a free bonus, the Greek root of stochastic. Oh, now we’ve got your attention…)
To 06.02 here in forum or here in wiki.