Question 1Suppose that we regress the hourly wage, W of each individual on the individual’s years ofexperience (Exper) and years of education (Educ). The estimated model is Wi = b1 b2Experi b3Educi b4Experi × Educi
. We find that b2 > 0, b3 > 0, b4
following statements is true?a. The estimated marginal effect of education on hourly wage is lower for individuals with moreyears of experience. The estimated marginal effect of experience on hourly wage is higher whenindividuals have more years of education.b. The estimated marginal effect of education on hourly wage is lower for individuals with moreyears of experience. The estimated marginal effect of experience on hourly wage is lower whenindividuals have more years of education.c. The estimated marginal effect of education on hourly wage is higher for individuals with moreyears of experience. The estimated marginal effect of experience on hourly wage is higher whenindividuals have more years of education.
About the Syllabus (1/2) Lecturer: Dr Bin Peng Emails:
[email protected],
[email protected] Office: Room 5.66, Level 5, Building H, Caulfield Campus Consultation Hours: Available HERE Assessment Requirements: I 1st Assignment: an empirical study, 20% I 2nd Assignment: an empirical study, 20% I Final Exam: 60% BP (Monash) 1 / 13
[email protected] [email protected] https://www.dropbox.com/s/789dy2n18yo8jxv/General%20Information.pdf?dl=0 About the Syllabus (2/2) Textbook: Hill, Griffiths and Lim (2018), Principles of Econometrics, 5th edition, Wiley and Sons. Read online — through library HERE Note that lectures only cover some key points of the relevant chapters. Reading lectures slides probably can guarantee a pass. Understanding all details of the book is the key to ensure HD. Reference Books: I Wooldridge (2012), Introductory Econometrics: A Modern Approach, 5th edition, Cengage Learning PDF version — free HERE Software: Eviews via the Monash online virtual environment MoVE. Tutorial 2 has detailed instruction. Also, a video is available under “Useful Material” of the Moodle. BP (Monash) 2 / 13 https://monash.hosted.exlibrisgroup.com/primo-explore/fulldisplay?docid=catau51440831880001751&context=L&vid=MONUI&lang=en_US&search_scope=au_everything&adaptor=Local%20Search%20Engine&isFrbr=true&tab=default_tab&query=any,contains,principles%20of%20econometrics&sortby=rank&facet=frbrgroupid,include,76415615&offset=0 https://economics.ut.ac.ir/documents/3030266/14100645/Jeffrey_M._Wooldridge_Introductory_Econometrics_A_Modern_Approach__2012.pdf Introductory Econometrics Topic 1: Introduction All we do in this unit is “Supervised Learning”. BP (Monash) 3 / 13 Why Econometrics? (1/2) BP (Monash) 4 / 13 Why Econometrics? (2/2) BP (Monash) 5 / 13 Econometrics: What is it? (1/2) Econometrics is a set of research tools used for the measurement and empirical testing of relationships in economic variables. The tools are from economic theory, mathematics and statistics. Varian, H. R. (2014), “Big Data: New Tricks for Econometrics.” Journal of Economic Perspectives, 28 (2): 3-28. Traditionally, the main aim of econometrics is to answer questions like - if one variable changes, by how much it will affect other variables. For example, • A public transportation council in Melbourne must decide on how an increase in tram/train/bus fares will affect the number of travellers who switch to car or bike. • A Melbourne city council ponders the question of how much violent crime will be reduced if an additional million dollars is spent on putting more police on the street. BP (Monash) 6 / 13 https://pubs.aeaweb.org/doi/pdf/10.1257/jep.28.2.3 https://pubs.aeaweb.org/doi/pdf/10.1257/jep.28.2.3 Econometrics: What is it? (2/2) • A CEO of OMO must estimate how much demand there will be in 10 years for the detergent OMO, and how much to invest in a new plant and equipment. Traditionally, it is concerned with quantifying relationship in the following ways: • Estimating the relationships between variables • Predicting/forecasting the behaviour of the variables. • Testing hypothesis involving economic behaviours It is also used by other disciplines like finance and marketing (e.g., A/B testing)... BP (Monash) 7 / 13 Economic Data Types To start using econometrics tools, we need to first collect the data, and use the information of the data to do the estimation. • Cross-sectional (CS) data • Time series (TS) data • Panel/longitudinal data Stock return of each trading day of companies Day 1 Day 2 Day 3 Company 1 0.01 -0.05 0.02 · · · TS data Company 2 0.05 0.08 -0.03 Company 3 -0.20 0.00 0.02 ... CS data Panel/longitudinal data BP (Monash) 8 / 13 Cross-sectional Data Each observation is a new individual unit that can be firm, geographical region, household, etc., with information at a point in time. A random sample from some underlying population - if the observation is not a random sample, we have a sample-selection problem. For example, survivorship bias. Usually, it is known as micro-data. BP (Monash) 9 / 13 https://en.wikipedia.org/wiki/Survivorship_bias Time Series Data Data on one variable collected over time. For example, macro variables (GDP, Interest Rate, Inflation Rate) reported in daily, monthly, quarterly or annually, or high frequency data which do not have equal time intervals. Usually it is known as macro-data. BP (Monash) 10 / 13 Panel/Longitudinal Data Panel data have observations on individual micro-units who are followed over time, i.e., combination of cross-sectional and time series data. Most part of this unit is concerned with the analysis of cross-sectional data, but many techniques learned throughout can be applied to time series and panel data studies. BP (Monash) 11 / 13 Data Types • Quantitative (Prices, Income, ...) • Qualitative (Gender, Marital Status, ...) In this unit the variable that we are interested in “explaining” is quantitative variable. We will use both quantitative and qualitative variables to explain it. BP (Monash) 12 / 13 Review of Statistical Concepts (in Week 1 Tutorial) • Random variables (p. 16) • The rules of summation (p. 22) • Expectation of a function of a random variable (p. 24) • The variance of a random variable (p. 26) • The variances and covariance of two random variables (p. 27) • Read also PoE appendix B (optional) BP (Monash) 13 / 13 Introductory Econometrics Topic 4: Prediction, Goodness-of-fit, and Modelling Issues BP (Monash) 1 / 28 Outline • Least Squares Prediction • Measuring Goodness-of-fit • Modelling Issues BP (Monash) 2 / 28 Least Squares Prediction (PoE 4.1) (1/2) • True Model: yi = β1 + β2xi + ei • Estimated Model: ŷi = b1 + b2xi Predict a value of y (call it y0) for a given value of x (call it x0), i.e., E[y0 |x0] = E[β1 + β2x0 + e0 |x0] = β1 + β2x0. (1) The point predictor is ŷ0 = b1 + b2x0, (2) which yields the forecast error f = y0 − ŷ0 = (β1 + β2x0 + e0)− (b1 + b2x0). (3) BP (Monash) 3 / 28 Least Squares Prediction (PoE 4.1) (2/2) It turns out that the prediction ŷ0 is “unbiased” in the sense that the mean forecast error is zero, i.e., E[f |x, x0] = E[(β1 + β2x0 + e0)− (b1 + b2x0) |x, x0] = β1 + β2x0 + E[e0 |x0]− (E[b1 |x] + E[b2 |x]x0) = β1 + β2x0 − (β1 + β2x0) = 0, (4) where x = {x1, . . . , xN} includes all observations of the regressor. Why conditional on x? ŷ0 is the Best Linear Unbiased Predictor (BLUP) for y0. Under the Machine Learning context, estimating the values of b1 and b2 is the so-called “training the model”. Prediction mentioned above is the so-called “testing”. Correspondingly, {yi, xi}Ni=1 used for regression are called the training set. (y0, x0) is the test set. BP (Monash) 4 / 28 Prediction Interval (1/2) The variance of the forecast error is Var[f |x, x0] = σ2 ( 1 + 1 N + (x0 − x)2∑N i=1(xi − x)2 ) , (5) where some of these terms have appeared in Var[bk |x]. Note that (x0 − x)2 measures how far x0 is from the centre of the x-value. Therefore, x0 far away from the centre indicates large forecast variance. Remark: 1. Large ∑N i=1(xi − x) 2 indicates small forecast error. 2. Small (x0 − x)2 indicates small forecast error. From the Machine Learning point of view, to be precise on prediction, you would like to have lots of training data, and the values of the test set are close to the center of training data. BP (Monash) 5 / 28 Prediction Interval (2/2) The estimated variance of the forecast error is V̂ar[f |x, x0] = σ̂2 ( 1 + 1 N + (x0 − x)2∑N i=1(xi − x)2 ) = σ̂2 ( 1 + 1 N ) + (x0 − x)2 · σ̂2∑N i=1(xi − x)2 = σ̂2 ( 1 + 1 N ) + (x0 − x)2 · V̂ar[b2 |x]. (6) Then the standard error of the forecast is se(f) = √ V̂ar[f |x, x0]. The 100 · (1− α)% prediction interval is [ŷ0 − tc · se(f), ŷ0 + tc · se(f)], (7) where tc = t(1−α/2,N−k), N is the sample size, and k is the number of parameters. Remark: Prediction may be improved if we (1). include more observations, (2). include more important variables apart from income in the model (Topic 5), or (3). improve our modelling technique (beyond this subject). BP (Monash) 6 / 28 Example (1/2) Figure 1: Regression Results of Food Expenditure Calculate the prediction interval for household with income $2000. By (7), we need three values: ŷ0, tc, se(f). (8) BP (Monash) 7 / 28 Example (2/2) 1. ŷ0 — Point estimation ŷ0 = b1 + b2x0 = 83.4160 + 10.2096 · 20 = 287.6089 (9) 2. se(f) — Standard error of the forecast V̂ar[f |x, x0] = σ̂2 ( 1 + 1 N ) + (x0 − x)2 · V̂ar[b2 |x] = (89.517)2 ( 1 + 1 40 ) + (20− 19.6048)2 · (2.0933)2 = 8214.31, (10) where the value of x is obtained from the data, and the others are read from Figure 1. Thus, se(f) = √ 8214.31 = 90.6328. 3. tc = t(0.975,38) = 2.0244, where 38 = 40− 2, 40 is the sample size, and 2 is the number of parameters. 4. The 95% prediction interval is [ŷ0 − tc · se(f), ŷ0 + tc · se(f)] = [104.1323, 471.0854]. (11) The 95% prediction interval suggests that a household with $2000 income will spend somewhere between $104.1323 and $471.0854 on food. BP (Monash) 8 / 28 Measuring Goodness-of-Fit (PoE 4.2) (1/3) Recall the simple linear regression model yi = β0 + β1xi + ei, (12) in which yi is made up of two components: • β0 + β1xi is the explainable or “systematic” component; • ei is the random, unsystematic and unexplained component. If the model is good, we would expect the explainable part to be as large as possible relative to the unexplained part. BP (Monash) 9 / 28 Measuring Goodness-of-Fit (PoE 4.2) (2/3) Note that using the estimated ŷi we can also write yi as yi = ŷi + êi = (b1 + b2xi) + êi, (13) where obviously êi = yi − ŷi. Subtract y from both sides to obtain yi − y = (ŷi − y) + êi, (14) where y = 1N ∑N i=1 yi. It can be shown that N∑ i=1 (yi − y)2︸ ︷︷ ︸ SST = N∑ i=1 (ŷi − y)2︸ ︷︷ ︸ SSR + N∑ i=1 ê2i︸ ︷︷ ︸ SSE , (15) where SST = Total Sum of Squares, SSR = Sum of Squares of Regression, SSE = Sum of Squares of Errors. BP (Monash) 10 / 28 Measuring Goodness-of-Fit (PoE 4.2) (3/3) SST = SSR + SSE, where • SST measures the total variation in yi’s about the sample mean; • SSR measures the part of the total variation that can be explained by the regression; • SSE measures the