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- United-kingdom
- University of Bristol
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- Ch6: The Linear Model

Harry W.

• 7

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The Linear Model

A linear statistical model is an equation between random variables:

Y = b_{0} + b_{1}X_{1} + ... + b_{K}X_{K} + U,

where:

- Y, X
_{1}, ..., X_{K}, and U are random variables - Y is the outcome, also called dependent variable.
- X
_{1}, ..., X_{K}are regressors, also called explanatory variables. - U is the error term.
- The parameters b
_{0}, b_{1}, ..., b_{K}are real numbers.

This model writes the outcome of interest Y (say the wage) as a linear function of some explanatory variables (say age, gender, education) plus an error term U (potentially an omitted variable, say IQ).

U collects everything in Y that is not b_{0} + Σ_{i=1}^{n}b_{k}X_{k}.

The b parameters can have a causal interpretation. The parameter b_{k} (k≥1) is the marginal effect of X_{k} on Y keeping all other regressors **and** the error term constant:

b_{k} = __∂Y___{ } |

∂X_{k}| X_{l≠k},U

Hence, keeping all X_{l} (l≠k) regressors and the error term U constant, we could look at how Y varies with X_{k} to know the causal effect of X_{k} on Y.

Exogeneity Assumption

The exogeneity assumption is a restriction on the joint distribution of the error term and the regressors:

E(U|X) = 0

Differently put, for any values x_{1}, ..., x_{K} we have:

E(U| X_{1} = x_{1}, ..., X_{K} = x_{K}) = 0.

The expected error for any value of the regressors is zero.

By the law of expected iterations, the exogeneity assumption implies:

- E(U|X
_{k}) = 0, for any k∈[1,k] - E(U) = 0

This then implies: cov(U,X_{k}) = 0.

Under the exogeneity assumption, we have:

E(Y| X_{1} = x_{1}, ..., X_{K} = x_{K}) = b_{0} + b_{1}X_{1} + ... + b_{K}X_{K}.

We can then study the causal effects of the X variables on the outcome without observing the error term U.

Assumptions of OLS

- Exogeneity
- i.i.d
- No Multicollinearity
**X**has full rank- No (very few) outliers.

Independent Identically Distributed (i.i.d) Assumption

We have an i.i.d sample with n realisations of Y and X: (y_{i}, x_{1i} ..., x_{Ki})_{i∈[1,n]}.

This assumption is needed to apply the Law of Large Numbers and the Central Limit Theorem, in order to show the properties of the OLS estimator.

With an i.i.d. sample, we also have n realisations of the error term u_{1}, ..., u_{n}, which are assumed to be independent.

No Multicollinearity

X_{1}, ..., X_{K} are not multicollinear: there is no linear relationship between any two or more than two, X-variables.

If this assumption does not hold, the b parameters in the linear model are not uniquely defined.

Also, the sample matrix **X** has full rank: there is no linear relationship between two, or more than two, columns of **X**.

This means that no regressor brings information that is redundant given the other regressors.

No Outliers

There are no, or not too many, large outliers.

OLS with Assumptions

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