Nature and Scope of Econometrics PDF

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Nature and Scope of Econometrics • Econometrics means ‘economic measurement’ • Econometrics attempts to measure quantitatively the concepts or assumptions developed by economic theory • And uses these measures to prove or disprove the economic concept or assumption

How do you do econometrics? 1) Creating a statement of theory or hypothesis: “As individuals income increase, their consumption will increase” 2) Representing this with a simple mathematical model (Consumption) = £100 + 1/2 (Income) 3) Specifying your statistical, or econometric model: C =

𝞪+𝞫x𝞘 +u

4) Collecting the data on consumption and income 5) Estimating the parameters

𝞪 and 𝞫

6) Checking for model adequacy ! a) Simple (two-variable) linear regression model C=

𝞪 + 𝞫𝞘 + u

!

!

!

b1) Multivariate linear regression model

!

!

C=

𝞪 + 𝞫₁𝞘 + 𝞫₂(interest rate) + u

7) Hypothesis testing - !Using the model to test hypothesis suggested by economic theory 8) Prediction or forecasting - What is the mean value of Y given X₁ and X₂ (income and interest rate)?

Classical linear regression model The method of Ordinary Least Squares (OLS): Two variable example

Criteria: To minimise the sum of the squared distances between the actual Y observations and (the fitted values) on the regression line

Basic Ideas of Linear Regression • Focus on the two variable model: !

Y=

𝞪 + 𝞫X + u

• Regression analysis studies the relation between Y and X - !Y: dependent or explained variable - !X: independent or explanatory variable - !𝞪 and 𝞫 are parameters

- !𝞪 is the intercept (coefficient) - !𝞫 is the slope (coefficient) Intercept and Slope

population Regression Function • Example: - !Y represents the math SAT score (US SAT exam ≈ A-Levels) - !X represents annual family income - !You have access to an hypothetical ‘population’ of 100 students that !

-

!

have taken the math SAT exam !Data are organised by income class and student

Population of 100 high-school students

• The round dots

connected with the line are the mean values for each income level • These points are called the conditional means or conditional expected values • The line connecting the conditional means is the population regression line (PRL)

• PRL: tells how the mean value of Y is in relation to each value of X • The PRL can be expressed as the population regression function (PRF):

• of X (=

is the mean or expected value of Y conditional upon a given value )

• What does it mean? It means on average, Y is equal to 𝞪 + 𝞫

population Regression Function (cont.) • How do we explain the score of an individual student in relation to income? - Any individuals maths SAT score is equal to the average for that group ± some quantity

Illustration

Two versions of PRF Deterministic PRF:

Stochastic PRF:

SAMPLE VS POPULATION • Usually we don’t have access to the whole population of students, but to a sample (i.e. sub section) of the population. Can we estimate the PRL from the sample data?

• Example: Suppose you’ve never seen table 2-1 above, the only data available to you are sample data shown in tables 2-2 and 2-3 below

Scatter plot of the two samples

• SRL: sample regression line • For each sample, we can draw a SRL: - SRL1: sample regression line drawn for sample 1 - SRL2: sample regression line drawn for sample 2 • For each SRL, we can define the sample regression function (SRF)

Sample Regression Function • Recall the deterministic PRF:

• The corresponding deterministic SRF:

• :estimator of • : estimator of 𝞪

, also called ‘fitted values’ of Y

• : estimator of 𝞫

• The stochastic PRF is:

• The corresponding stochastic SRF: -

: estimator of

, called ‘residuals’

How can we choose the best - fitting SRF to approximate PRF? PRL and SRL

• Consider the actual observation - In terms of SRF:

- In terms of PRF:

• To the left of A,

• To the right of A,

:

Least squares procedure Least squares principle: • To choose and small as possible:

such that the residual sum of squares

How does the OLS estimator work?

• and are called OLS or least squares estimators • These estimators minimise the sum of the squared (vertical) distances between the observations ( ) and the sample regression line • Y: miles per gallon, X: weight of a vehicle in kg

• For each value of X, we can predict a val • All predictions lie on the sample regressio

• The difference between the actual value o called residual:

is as

• Solution for the intercept:

• Solution for the slope coefficient:

EXAMPLE: REGRESSION LINE Consider a company producing tables (Y) employing workers (X). If the management decides to employ 25 workers, estimate the mean number of tables produced

• Using the data, we compute:

• Sample regression line:

• INTERPRETATION: 25 workers are expected to produce on average about 51 tables

Properties of OLS estimators • and are random variables • They are a function of Y which is also a random variable • Thus, and follow a sampling distribution • and are unbiased estimators of the population parameters 𝞪 and 𝞫

• Under additional assumptions, OLS estimators have the minimum variance of all unbiased linear estimators (i.e. most efficient among the group of unbiased and linear estimators) • OLS estimators as defined as BLUE (Best Linear Unbiased Estimator)

interpretation of the old estimators • Using the data in table 2-2 and running an OLS regression:

Interpretation • Slope: - If the family income ↑ by £1, the mean maths SAT score ↑ by 0.0013 points - Or if the family income ↑ by £1,000, the mean math SAT score ↑ by (0.0013 x 1000 =) 1.3 points

• Intercept: - If the family income is zero, the average math score is 432.4 (but income -

is never zero) Often the intercept has no economic meaning

Fitted Line Plot

‘Linear’ regression • Linearity in the variables (Xs):

• But these models are not:

• Linearity in the parameters (i.e. 𝞫s enter with the power of 1)

• Linear regression means regression linear in the parameters - It may or may not be linear in the Xs

causation VS correlation • Regression does not imply causation

• Causation: a cause (X) → an effect (Y) • Correlation: Y ⟷ X • Causation must be justified by the economic theory that you are testing, not by how you specify your econometric model...