“Suppose that (Y = alpha + beta X + U).” A sentence like that is certain to come back up dozens of occasions in an introductory econometrics course, but when I had my means it will be stamped out utterly. With out additional clarification, this sentence might imply any variety of various things. Even with clarification, it’s a supply of limitless confusion for starting college students. What’s (U) precisely? What’s the which means of “(=)” on this context? We are able to do higher. Listed below are just a few options.
Inhabitants Linear Regression
Typically (Y = alpha + beta X + U) is nothing greater than the inhabitants linear regression mannequin. In different phrases ((alpha, beta)) are the options to
[
min_{alpha, beta} mathbb{E}[(Y – alpha – beta X)^2].
]
The standard strategy to sign that is by including the “assumptions” that (mathbb{E}[XU] = mathbb{E}[U] = 0). It’s no surprise that college students discover this complicated. Neither of those equalities is actually an assumption; every is true by building. Relatively than “let (Y = alpha + beta X + U),” I recommend
Outline (U equiv Y – (alpha + beta X)) the place (alpha) and (beta) are the slope and intercept from a inhabitants linear regression of (Y) on (X).
This makes it clear that (U) has no lifetime of its personal; it’s outlined by the coefficients (alpha) and (beta). On this means, the equalities (mathbb{E}[XU] = mathbb{E}[U] = 0) develop into a theorem to be deduced slightly than a spurious “assumption” of linear regression. Repeat after me: the inhabitants linear regression mannequin has no assumptions. We are able to all the time select (alpha) and (beta) to make sure that (U) satisfies the equalities from above. The answer to the inhabitants least squares drawback is
[
beta = text{Cov}(X,Y)/text{Var}(X),quad
alpha = mathbb{E}[Y] – beta mathbb{E}[X].
]
By the linearity of expectation, it follows that
[
mathbb{E}[U] = mathbb{E}[Y – alpha – beta X] = mathbb{E}[Y] – (mathbb{E}[Y] – beta mathbb{E}[X]) – beta mathbb{E}[X] = 0
]
and equally, though with a bit extra algebra
[
begin{align}
mathbb{E}[XU] &= mathbb{E}[X(Y – alpha – beta X)]
&= mathbb{E}[X(Y – left{mathbb{E}(Y) – beta mathbb{E}(X)right} – beta X)]
&= mathbb{E}[Xleft{Y – mathbb{E}(Y) right}] – beta mathbb{E}[Xleft{X – mathbb{E}(X)right} ]
&= textual content{Cov}(X,Y) – beta textual content{Var}(X) = 0.
finish{align}
]
Conditional Imply Perform
In different conditions (Y = alpha + beta X + U) is meant to signify a conditional imply operate. That is often signaled by the belief (mathbb{E}[U|X] = 0). This time round I haven’t written the phrase assumption in “scare quotes.” That’s as a result of there’s an assumption lurking right here, in contrast to within the inhabitants linear regression mannequin from above. Nonetheless, it is a hopelessly complicated means of indicating it. Right here’s a greater means:
Outline (U equiv Y – mathbb{E}(Y|X)) and assume that (mathbb{E}(Y|X) = alpha + beta X).
Once more, this makes it clear that (U) has no lifetime of its personal. It’s constructed from (Y) and (X). The conditional imply operate (mathbb{E}(Y|X)) is solely the minimizer of (mathbb{E}[left{ Y – f(X)right}^2]) over all (well-behaved) features. By building (mathbb{E}[U|X] = 0) since
[
mathbb{E}[U|X] = mathbb{E}[Y – mathbb{E}(Y|X)|X] = mathbb{E}[Y|X] – mathbb{E}[Y|X] = 0
]
by the linearity of conditional expectation and the truth that (mathbb{E}(Y|X)) is a operate of (X). However how can we ensure that the conditional imply operate is linear? This can be a bona fide assumption: it might be true or it might be false. Both means, it’s a lot clearer to emphasise that we’re making an assumption in regards to the type of the conditional imply operate, not an assumption in regards to the error time period (U equiv Y – mathbb{E}(Y|X)).
Causal Mannequin
Each interpretations of (Y = alpha + beta X + U) from above are purely predictive; they are saying nothing about whether or not (X) causes (Y). To point {that a} linear mannequin is imply to be causal, it’s conventional to write down one thing like “suppose that (Y = alpha + beta X + U) the place (X) could also be endogenous.” Typically “could also be endogenous” is changed by “the place (X) could also be correlated with (U).” What on earth is that this speculated to imply? The language is obscure, evasive, and imprecise. It additionally stretches the which means of “(=)” past all motive. Right here’s my advised enchancment:
Contemplate the causal mannequin (Y leftarrow (alpha + beta X + U)) the place (U) is unobserved and ((X,U)) could also be dependent.
Causality is intrinsically directional: cigarettes trigger lung most cancers; lung most cancers doesn’t trigger cigarettes. The notation “(leftarrow)” makes this clear. In stark distinction, the notion of mathematical equality is symmetric. If (Y = alpha + beta X + U), it’s simply as true to say that (X = (Y – alpha – U) / beta). After all that is nonsensical when utilized to cigarettes and most cancers.
In a causal mannequin, (U) does have a lifetime of its personal; it represents the causes of (Y) that we can not observe. Maybe (Y) is wage, (X) is years of education and (U) is “household background” plus “potential.” For that reason I do not write “outline (U equiv (textual content{one thing})).” We aren’t defining a residual in a prediction drawback. We’re taking a stand on how the world works by writing down a selected causal mannequin. In a randomized managed trial, any unobserved causes (U) could be impartial of (X). Right here we now have not made this assumption. We now have, nonetheless, assumed a selected type for the causal relationship: linear with fixed coefficients. Every extra yr of education causes the identical improve (or lower) in wage no matter who you’re or what number of years of education you have already got. This mannequin might be fallacious. However proper or fallacious, it’s basically distinct from the inhabitants linear regression and conditional imply fashions described above. Let’s endeavour to make this clear in our notation.
