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Simple regression Statistics for dummies Statistics Gabriel V. Montes-Rojas Gabriel Montes-Rojas Statistics Simple regression Statistics for dummies y = β0 + β1x + u Much of applied econometrics is concerned with the linear simple regression model that explains the relationship between y and x: y = β0 + β1 x + u where y x dependent variable independent variable explained variable explanatory variable response variable control variable regressand regressor or covariate u is called the error term, residual or disturbance and represents all other factors, diﬀerent from x that aﬀect y . Gabriel Montes-Rojas Statistics Simple regression Statistics for dummies y = β0 + β1x + u Our interest is the eﬀect of x on the variable y on some population. The error term, u is assumed to have no systematic inﬂuence on y and therefore, only x is of importance. Then, we believe that y ≡ f (x ) = β 0 + β 1 x. The following deﬁnitions will be used extensively during the course: β 0 is the intercept, f (0) = β 0 . This represents the value of y when x is set at 0. β 1 is the slope, ∆y = β 1 . ∆x This represents the unit change in y after a unit change in x. Gabriel Montes-Rojas Statistics Simple regression Statistics for dummies Gabriel Montes-Rojas Statistics Simple regression Statistics for dummies Example 2.7 (p.41 in Wooldridge): Returns to education wage = β 0 + β 1 educ + u Wages are expected to be an increasing function of education, i.e. more education means on average higher wages. Then, in this linear model, we expect that β 1 > 0. What does u mean? Other factors, diﬀerent from education, that aﬀect wages, such as age or ability. Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model Statistics for dummies Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model Random variables (RV) Why do we need random variables in Econometrics???? We will (almost) never observe the whole population, only a small portion of it A random sample is a subset of a population If we consider the random variable X , a random sample is {xi }n=1 or x1 , x2 , ..., xn that consists of n realisations of the i variable X , which are indexed by i. Example: If X is the return of an asset, a random sample are actual observations in the market about the asset returns. Say for a sample of three observations x1 = $ 1000, x2 = −$ 567, x3 = $ 0 Example: Flipping a coin: let X = 0 be HEADS and X = 1 be TAILS. Then, X = {0, 1}. Moreover, P [X = 0] = P [X = 1] = 0.5. (This is called the Bernoulli distribution). Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model Discrete vs Continuous RVs A discrete random variable is one that takes on only a ﬁnite or countably inﬁnite number of values. Example: Flipping a coin: let X = 0 be HEADS and X = 1 be TAILS. Two possible values: 0 or 1. Example: Number of £50 bills in your wallet: X can take any number in 0, 1, 2, 3,..., ∞ Each outcome of X has an associated probability. pj = P (X = xj ), j = 1, ..., k. This probability measure satisﬁes: pj ≥ 0, j = 1, 2, ..., k ∑k=1 pj = 1 j Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model Discrete vs Continuous RVs A continuous random variable is one that takes on any real value. Let X be a continuous random variable. Its probability measure is described by a density function f (X ) that satisﬁes f (x ) ≥ 0 for all x ∈ X , where X is the domain of X , usually X =R X f (x )dx = 1 Although the density function acts as a probability of each value of x, it has a tricky interpretation, because there are so many values in X , that individually each one has probability zero (?!). Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model Expectation of a RV Random variables can be described by some of its features: Expectation: E [X ] What value should we expect from X ? If we have a considerable amount of draws from the X random variable, what would be their average? For the coin example: E [X ] = 0 × P [X = 0] + 1 × P [X = 1] = 0 × 0.5 + 1 × 0.5 = 0.5. For the discrete RVs: E [X ] = ∑k=1 xj × P [X = xj ]. j For the continuous RVs: E [X ] = X xf (x )dx. Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model Property of expectation: Let A and B be two random variables, and c and d two constants. Then, E [cA + dB ] = cE [A] + dE [B ]. Property of expectation: Let A and B be two independent random variables. Then, E [A × B ] = E [A] × E [B ]. Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model An estimator of the expectation of a random variable X is the sample average. Given a random sample {xi }n=1 , deﬁne x = n−1 ∑n=1 xi which is i ¯ i simply the average. ˆ ˆ An estimator µ is unbiased for a given parameter µ if E (µ) = µ In words, if we consider all possible random samples, on average, we will obtain the parameter we want to estimate. In our case, we can prove that E (x ) = E (X ). ¯ Proof:... Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model Variance of a RV However, for a given realisation of X , deﬁned as x, we may have that x = E [X ]. But, how much does this random variable deviate from the E [X ]? Variance: Var [X ] ≡ E [(X − E [X ])2 ] Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model Prove that Var [X ] = E [X 2 ] − (E [X ])2 . Property of variance: Var [aX ] = a2 × Var [X ] Property of variance: Var [aX + bY ] = a2 × Var [X ] + b 2 × Var [Y ] + 2ab × Cov [X , Y ], where Cov [X , Y ] = E [XY ] − E [X ]E [Y ] Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model Covariance The covariance of the random variables A and B measures how much co-movement they have. Covariance: Cov [Y , X ] ≡ E [YX ] − E [Y ]E [X ] Property of covariance: Let A and B be two independent random variables. Then, Cov [A, B ] = 0. Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model In the simple regression model... In the simple regression model, Y , X and U are random variables. β 0 and β 1 are population parameters, i.e. constants that describe the relation between Y and X . Then, E [Y ] = E [ β 0 + β 1 X + U ] = β 0 + β 1 E [X ] + E [U ] (Since U captures other factors, we will assume that E [U ] = 0.) However, our main interest is in the conditional expectation that deﬁnes the population regression model: E [Y |X ] = E [ β 0 + β 1 X + U |X ] = β 0 + β 1 X + E [U |X ] = β 0 + β 1 X Assumption: U and X are independent, then E [U |X ] = E [U ] = 0. Gabriel Montes-Rojas Statistics Expectation Simple regression Variance Statistics for dummies Regression model Parameters vs Estimators Note: β 0 and β 1 are population parameters to be estimated. ˆ ˆ β 0 and β 1 will be their estimators. The parameters are just numbers, they are ﬁxed. However, the estimators will be random variables. Gabriel Montes-Rojas Statistics

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posted: | 2/5/2011 |

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