# Identifying assumption meaning

I am hoping to get a good explanation regarding what is meant by an identifying assumption.

In many articles, under empirical strategy, authors state that:

• we exploit firm level variation to identify the effect
• we exploit industry level variation to identify the effect

Examples:

The paper examines the effect of bank runs on lending. We exploit variation in the structure of banks' liabilities to identify banks that were more vulnerable to the run.

We exploit the variation across geographically distinct commercial real estate markets to establish conclusively that loan supply shocks emanating from Japan had real effects on economic activity in the United States.

"Identification" is the most loaded term in econometrics. There are multiple cheap talk equilibria with regard to its meaning. It is used with different intended (but related and overlapping) meanings, in different contexts, by people with different orientations, with different levels of precision.

Therefore you will get a range of correct answers. Here is an attempt covering a few of the variations, going from the theoretical end of the spectrum to the empirical.

## Statistics

A statistical model is a one-to-one mapping $$\theta \mapsto P_{\theta}$$ from a given parameter space to a family of probability measures. It is the one-to-one property of the mapping that makes the model "identified". No two different elements in the parameter space can give rise to observationally equivalent data generating processes.

In statistics, a model is therefore, by definition/assumption, always identified. (This can be seen in the assumptions for all the foundational results, e.g. Neyman-Pearson.) Statisticians never speak about identification, because they don't have to.

For example, for $$y = \beta x + \epsilon \quad (*)$$ where $$(x,\epsilon)$$ is bivariate normal, to specify a model for population $$(x,y)$$ parametrized by $$\beta$$, one must assume that $$Cov(x, \epsilon) = 0$$. Without imposing this assumption, different $$\beta$$'s could give rise to the same distribution for $$(x,y)$$. In econometrics, which is much more explicit about the identification issue, the condition $$Cov(x, \epsilon) = 0$$ will sometimes be called an identification assumption.

## Structural Models

If one tries to construct a statistical model by adding unobserved disturbances to an economic model, identification needs to be addressed. In order to make the resulting structural econometric model identified, usually one needs to make certain assumptions, either of economic or technical nature. These are called identification assumptions.

For example, suppose there are $$n$$ firms in Cournot competition with private constant marginal costs $$(c_1, \cdots, c_n)$$ drawn from joint density $$f(x_1, \cdots, x_n)$$. The econometrician observes firms' output $$(q_1, \cdots, q_n)$$ and market price $$P$$ and would like to identify $$f$$. One possible identification assumption is that the Jacobian of the FOC system $$\frac{d P(Q)}{dQ} q_i + P(Q) - c_i = 0, \, i = 1, \cdots, n,\, \mbox{ where } Q=\sum_1^n q_i$$ is non-vanishing. Then, by the Implicit Function Theorem, $$(q_1, \cdots, q_n)$$ maps one-to-one locally to $$(c_1, \cdots, c_n)$$. This implies the model, parametrized by the observed quantity $$(q_1, \cdots, q_n)$$, is identified, at least locally. The empirical interpretation is that sufficient variation in the trade-offs faced by the firms allows you to identify $$f$$.

There are more interesting examples where the identification assumption puts restriction on economic agent behavior, etc.

## Empirical Usage-Consistent Estimation

So far, identification is purely a property of the mapping from parameter to data generating processes. Identification is a prerequisite for estimation but by itself it makes no mention of the sample.

There are also contexts where an econometrician speaks about a specific estimator that's designed to estimate a specific parameter in a specific model. An assumption under which the estimator consistently estimates the parameter is called an identification assumption. For example, given time-series data $$(x_t, y_t)$$ generated by $$y_t = \beta x_t + \epsilon_t, \; t = 1, 2, \cdots, \quad (**)$$ the parameter $$\beta$$ "can be identified by OLS $$\hat{\beta}$$" under the assumption that $$Cov(x, \epsilon) = 0$$.

In $$(*)$$ and $$(**)$$, the condition $$Cov(x, \epsilon) = 0$$ and the terminology are the same, but "identification assumption" have different (but clearly related) meanings.

## Empirical Usage-Causal Inference

When one is interested in establishing causal effect, a condition imposed on the model that allows for causal interpretation of the estimate is called an identification assumption. Yes---$$Cov(x, \epsilon) = 0$$ for the linear model would fall under this category also. Often it is strengthened to $$E[\epsilon|x] = 0$$, which is more interpretable for causal inference.

Similarly, when $$Z$$ is an instrument, the exogeneity condition $$Cov(Z, \epsilon) = 0$$ is an identification assumption. For diff-in-diff, the parallel-trends condition is an identification assumption. For regression discontinuity design, the identification assumptions are that, first, there are no other discontinuities except the forcing variable, and second, agents cannot manipulate the forcing variable. The corresponding empirical design (e.g. IV/DID/RDD/etc) is sometimes called the identification strategy.

In this context, "identification" is not a binary condition. One could have weak identification, e.g. a weak instrument.

Used in this sense, an identification assumption clearly needs to be justified when it's claimed to hold empirically. In other words, one needs to justify that the corresponding variation is exogenous---e.g. the variation of the instrument is exogenous, etc.

The paper examines the effect of bank runs on lending. We exploit variation in the structure of banks' liabilities to identify banks that were more vulnerable to the run...

vulnerability to a run is clearly a endogenous variable in relation to lending. The claim is then the empirical design in question uses exogenous variation in the structure of banks' liabilities---as an instrument/forcing variable/whatever---to circumvent endogeneity and achieve identification.

I think the best way how to explain this is to first quickly explain what identification actually is. As mentioned in this thread:

For example, in the John Stachurski "A Primer in Econometric Theory" the identification is a process of finding out if the parameters are identifiable and identifiability is defined as

“Identifiability means that the parameter vector associated with the unknown distribution can eventually be distinguished from the data.”

Furthermore, as nicely summarized by BBKing:

An identified estimate is any estimate that fulfills certain conditions that make it the true number we want.

For example, any coefficients from (estimating) an OLS regression are estimates. However, a coefficient from an OLS model that fulfills all the OLS assumptions for an unbiased consistent estimate (e.g. no relationship between the error terms and the independent variable) is an identified estimate. Only such a model "achieves identification" or allows authors to state "we identify the effect".

So what are identifying assumptions? They are assumptions under which it is possible to say that the parameters are identifiable. For example, in simple OLS

$$y = X \beta +e$$

a condition for a parameters to be identifiable is that $$X’X$$ matrix, which is used to estimate the $$\beta$$ (since $$\hat{\beta} = (X'X)^{−1}X'y$$), must be invertible. If the matrix $$X'X$$ is not invertible we simply cannot identify the model parameters.

Different models will have different conditions for parameters to be identifiable. You might want to consult some econometric textbook such as Verbeek's guide to modern econometrics or Pesaran Time Series and Panel Data Econometrics or some undergraduate textbooks such as Wooldridge introduction to modern econometrics if you consider the previous suggestions to be too advanced for your level to know what identification conditions are for different models (although undergraduate textbooks will only mention those very briefly).

• The invertibility of $X'X$ is a very narrow sample-by-sample "identification assumption". According to this condition, for some samples identification holds and for others it doesn't. Probably more sensible to have a population condition along the lines of "eigenvalues of E[X'X] is bounded below"---under which the probability of $\frac{1}{n}X'X$ being invertible approaches 1 as sample size get large. Aug 22 '20 at 5:02
• @Michael yes but I just wanted to provide a concrete and simple example for the user who looks from the way question is written looks like undergraduate
– 1muflon1
Aug 22 '20 at 8:55

"Identification" is the professional jargon in econometrics for "asserting that the outputs from an econometric model do indeed estimate what we want and declare that they estimate".

"Identification" does not include an assertion that a specific estimate coming from combining a specific estimation method with a data sample, will be a "good" estimate (unbiased, consistent, etc). It only asserts that we can, somehow, in principle estimate what we declare that we want to estimate. This leads to various conditions, sometimes technical, that cannot be summarized.

But in the examples that the OP sites, "identification" is not used in this sense but rather in a much more general and informal way, because (exploiting the) "variation", is at the very heart of statistical estimation: the only way to separate / estimate effects is if there is some varying relationship between ... varying variables.

Consider for example wage as the dependent variable and years of education as the explanatory variable: if we observe variation in the wage in a sample where all workers have the same years of education, we cannot estimate what, if any, is the effect of years of education on the wage. Reversely, assume that we have a sample where all wages are the same, but years of education vary per worker. Here we could say that years of education do not matter for the wage, but still, this will only be a sample-specific conclusion, we could not generalize it. But if both wage and years and education vary, then we could in principle estimate their relation, which could also be zero.

In other words, to estimate a structural (i.e. fixed, permanent in some sense) effect, we need variation (difference, change): one of the many stimulating (apparent) contradictions that characterize statistics.