# What is parameter identification in the context of OLS?

Can someone explain what identification means in the context of an OLS model? I have a fair grasp of the derivation using either the method of moments or by minimizing the squares, but am failing to grasp which part of this process corresponds to identification. Also, how does identification differ from estimation of the parameters?

• In which context did you encounter this term? Jun 8, 2016 at 20:49

Thank you for all the responses. It has been more than a year since I asked this question and I am now able to provide one answer to the question. The below answer illustrates the issue of identification in the context of treatment evaluation where the parameter of interest is the causal effect of treatment receipt on some outcome of interest. Such evaluation problems arise frequently when you are trying to estimate the efficacy of a drug by comparing health outcomes of those who received treatment against a control group. This problem is also frequently encountered in the social sciences where you might be interested in estimating the treatment effect (causal effect) of some policy intervention (e.g., subsidizing healthcare for a certain group of people) on various outcomes such as income, mortality, etc.

Setup:

For simplicity, let the true underlying process be given by the following linear relationship: $$y=\alpha_0+\alpha_1 t + \bf{Z}\pmb{\beta}+\varepsilon$$ where $y$ is the outcome, $t$ is a binary indicator for receipt of some treatment of interest, and $\bf{Z}$ is a vector of all relevant factors that affect the outcome $y$. Further, let $\varepsilon$ be a normally distributed mean-zero noise term (this noise is assumed to be truly non-deterministic since we have assumed $\bf{Z}$ contains every relevant determinant of $y$). It follows that $$\alpha_1=E[y | t=1, \textbf{Z}]-E[y | t=0, \textbf{Z}]$$ Since $\bf{Z}$ includes all relevant factors that determine the outcome $y$, we interpret $\alpha_1$ as the causal effect of treatment receipt ($t=1$) on the outcome $y$.

Empirical Application:

Suppose we are given data on $y$, $t$, and $\bf{Z'}$, where $\bf{Z'}$ is a subset of $\bf{Z}$. In other words, $\bf{Z'}$ only contains some of the relevant variables that determine the outcome $y$. Let $\bf{\tilde{Z}}$ be the unobserved components of $\bf{Z}$. We need to estimate the causal effect of receiving treatment, i.e., $\alpha_1$, in our empirical exercise. However, given our inability to observe the full vector $\bf{Z}$, the best we can do with OLS is to estimate the following: \begin{align} y&=\hat{\alpha}_0+\hat{\alpha}_1 t + \bf{Z'}\hat{\pmb{\beta}}+\nu \end{align} where the error term $\nu = \varepsilon+\bf{\tilde{Z}}\pmb{\tilde{\beta}}$ absorbs the effect of $\bf{\tilde{Z}}$ and is thus, no longer random noise.

Now notice that \begin{align} E[y | t=1, \textbf{Z}']-E[y | t=0, \textbf{Z}'] &=\hat{\alpha}_1 + \left(E[\nu | t=1, \textbf{Z}']-E[\nu | t=0, \textbf{Z}']\right) \\ & = \hat{\alpha}_1 + \pmb{\tilde{\beta}}\underbrace{\left(E[\bf{\tilde{Z}} | t=1, \textbf{Z}']-E[\bf{\tilde{Z}} | t=0, \textbf{Z}']\right)}_{bias} \\ \implies &\hat{\alpha}_1= \left(E[y | t=1, \textbf{Z}']-E[y | t=0, \textbf{Z}']\right) - \text{bias} \end{align} Therefore, $\hat{\alpha}_1$ captures both the mean difference in outcomes associated with treatment status as well as a bias stemming from heterogeneity in $\bf{\tilde{Z}}$ w.r.t. treatment status.

A simple OLS doesn't allow us to adjust for such biases. Therefore, $\hat{\alpha}_1$ does not provide the causal effect of $t$ on $y$. We have thus, failed to identify our parameter of interest namely, the causal effect of $t$ on $y$. The OLS estimate $\hat{\alpha}_1$ can only be interpreted as an estimate of the correlation between $y$ and $t$ that adjusts for $\bf{Z}'$.

Conclusion:

The above explanation began by defining the parameter of interest as the causal effect of $t$ on $y$. It then illustrated how identification of this parameter can be compromised by omitted variable bias. There are a number of empirical strategies that can be employed to correct for such biases. The most obvious would be to collect data on the missing variables $\bf{\tilde{Z}}$, however, this is unlikely to be feasible. Another option would be to randomize the assignment of $t$ and then assume that $t$ is independent of $\bf{\tilde{Z}}$ by construction. This works well if you have the time and resources to design your own intervention $t$ and collect outcomes data on the subjects of your study. If you are stuck with observational data that cannot be randomized, you will need to look for empirical strategies that allow you to induce exogeneity in the assignment of $t$. Social scientists refer to such approaches as quasi-experimental methods.

Identification means that the mapping from parameter values of interest to the distribution of observational data is injective, hence the parameter values of interest can be uniquely inferred with full knowledge of the distribution of data.

• That definition more closely resembles "consistency" in terms of estimation, that is that an estimator goes from $\hat{\beta} \rightarrow \beta$ as $n \rightarrow \infty$. Is it possible that expressing $\beta$ as a function of the joint density $f_{x,y}$ is this aspect of "identification"? Jan 3, 2018 at 19:48
• @AdamO Good point. I've changed my language somewhat. Jan 3, 2018 at 21:32

it means that for your model two parameter estimates are somehow correlated with each other or one estimate is a linear combination of several other. This might be best understood in the context of contrast coding to test hypothesis in a meaningful manner. Suppose you had two groups that you wanted to compare, say a1 and a2, one comparison would be to consider the differences of a1-a2..this model is identifiable as you have two groups and you will fit two parameters. Another way to look at this question would be to say what the increase/decrease from a global mean . that is a1=a+b1 and a2= a+b2 .. now you have 3 parameters in this case a, b1 and b2 and the parameters are non-identifiable, as one parameter is redundant. However you can always fit the model using a linear combination that makes the model identifiable. This may not be the most clear answer and I will be glad if someone can either edit this or post an answer that better. There is also a much better answer on this stack overflow post What is model identifiability?

• This post is very unclear, and borderline wrong. Identification follows from a strict exogeneity assumption, it does not,require a fully or overspecified model Jun 8, 2016 at 19:21
• I will be glad to remove it if its wrong. Can you please post a more clear answer then and I will delete my answer Jun 8, 2016 at 19:36
• from the Wiki definition of identifiability "A model that fails to be identifiable is said to be non-identifiable or unidentifiable; two or more parametrizations are observationally equivalent. " which i what i wanted to explain above Jun 8, 2016 at 19:38
• @Repmat - a multicolliearity issue also is a version of an identifiability issue, and yet it does not seem directly related to a strict exogeneity condition $E(\epsilon|X)=0$, no? Jun 10, 2016 at 12:08
• @christophhanck yeah sure, but I have yet to encounter a model where MC was problem for identification. Seldom does perfect correlation exists in the population (expert for various dummy traps) Jun 10, 2016 at 13:01