Fixed Effects Using Least Squares Dummy Variable Model

[Margarita Lovvorn]

Away to confirm this result is to collect data about each country over an extended period of time. In this way, we move from cross-sectional data to panel data, where each variable in the datset is observed along a given time period.

However, if we change the way we look at the data and consider the relationship between international aid and growth within one single country and across time, we would observe a positive relationship.

One possible explanation is that country C is poorer and, because of that, it receives more funding than the other countries. However, when we compare country C against other countries, its growth is still limited. As results, across countries variation is negative. The trend is positive only if we compare growth over time within each country. As country C receives more funding, its economic growth improves.

The OLS results show that the coefficient of International Aid is significantly and negatively correlated with Economic Growth. It suggests that for each additional milion received in international aid funding, economic growth decreases by 11%.

We compare results across a cross-sectional OLS and a pooled OLS model. The pooled OLS regression indicates that international aid has no effect on economic growth; the beta coefficient is not significantly different from zero.

As we discussed before, OLS can be problematic with panel data because it takes into account across-country variation. The effect of international aid on growth is negative because the overall trend across countries is negative. Countries which receive more funding (like Country C) report a lower economic growth. To observe the true effect of international aid, we need to remove the across-country variation. We can do it by demeaning the data.

De-meaning requires you to substract the mean from each variable. In this case, we are interested in demeaning by group, so that the values of x and y are centered around each group mean. You can observe the process in table 1.5.

The mean value of Economic Growth in group A is 4. We substract 4 from each value of Economic Growth within group A (2, 4, 6) and obtain -2, 0, and 2. We repeat the same process for each group, and for both Economic Growth (x) and International Aid (note: you need to demean all the variables in your model, including control variables).

As a result, we obtain a new dataset where our observations are disposed along a line. In graph 1.10, there are no longer significant differences across the 3 countries - they all lay on the same line. We observe only within countries differences.

A fixed effect model is an OLS model including a set of dummy variables for each group in your dataset. In our case, we need to include 3 dummy variable - one for each country. The model automatically excludes one to avoid multicollinearity problems.

Results for our policy variable in the fixed effect model are identical to the de-meaned OLS. The international aid coefficient represents a one-unit change in the economic growth, on average, within a country for each additional one-milion dollar received.

The fixed effect model also includes different intercepts for each country. The intercept for country A is represented by the constant. It represents the average economic growth when international aid is equal to 0. Country B intercept is 4.90 points percentage lower than Country A. Country C intercept is 12.8 points percentage lower than Country A. The coefficient of Country B and Country C indicate how much a country is different, on average, from country A (the reference category).

We plot our results in graph 1.12. As you can see, the lines have the same slope, but different intercept. The slope is represented by the coefficient of international aid, the policy variable. It represents the average within-group variation across time. The intercepts represent the average of each group when international aid is equal to 0. A fixed effect model better approximates the actual strucutre of the data and controls for group-level characteristics.

Data are structured in a panel dataset where we have 10 companies and observations over a five-year period. You can see here the structure of the data for Company 1, Company 10, and Company 2.

Each company has different levels of investment in research and development. We plot each company average R&D investement in graph 2.2. Each bar represents a company. The middle dot is the average R&D investment. The bars go from the minimum level of investment to the maximum.

You can see how Company 1 has a lower average R&D investment than Company 2. Company 5 has a smaller range than the other companies, with a minimum investment that it is higher than the average outcome of company 6 and so on.

There are can be several explanation for these differences: size, different managerial practices, or business culture, among others. Note that this variable changes across organizations but not across time.

Variation across years can be explained by new national policies that get implemented, internaional agreeements, taation policies, or the economic situation. Note that these varibles vary across years but not across entities (e.g., national policies affect all companies in the dataset).

As in the model with the dummy variables, the coefficient of the fixed effect model indicates, on average, how much the outcome (Company R&D) changes per country over time for a one unit increases of x (Public R&D).

The plm package also allows us to test whether the fixed effect model is a better choice compared to the OLS model with the \(pFtest\) command. If the p-value is lower than 0.05, than the fixed effect model is a better choice.

As replication study we are going to look at research on deterrence theory. Deterrence theory argues that individuals refrain from engaging in deviant activities if they know they will be punished and if they know the pain will be severe, as illustrated in figure 3.1. This theory is often applied to justify more severe penalties ( such as the death penalty ) and explain why crime rates have decreased in recent years. Of course, there are mixed opinions and evidence.

The average crime rate across all counties and years is 3.84%. The average probability of arrest is 30.74%. The average probability of convinction is 68.86%. The average probability of prison sentence is 42.55%. The average length of prison stay is 8.955 days.

Interpretation of the coefficients is tricky since they include both the within-entity and between-entity effects. It represents the average effect of X over Y, when X changes across time and between countries by one unit.

There are tests that you can run to chose between fixed and random effects. For instance, the Hausmann test compare a fixed and a random effect model. If the p-value is I have a question about if there is a substantive difference between a fixed effect and the way we estimate them (e.g., dummy variables). Are the estimated dummy variables the fixed effect, or do they simply absorb the fixed effect (and other variables invariant across the other dimensions of the data)?

So in this specification, we aren't actually estimating the true fixed effect, we are instead using dummy variables to estimate the $\hat\lambda_i$'s as a way to absorb time invariant components of $y$ for each $i$? And if this understanding is correct, then is the second equation the more accurate way to write down what we are estimating in practice?

To be clear, estimating your equation via least squares dummy variables (LSDV) is algebraically equivalent to estimation in deviations from means. Put differently, including indicator variables for all $N-1$ entities in your panel produces mathematically equivalent estimates of $\beta$ to those where you run ordinary least squares on the 'time demeaned' data.

Suppose you wanted to investigate the effect of unemployment on vehicle thefts at the city level. You sample five cities and observe general rates of auto larceny over two time periods (i.e., 2019 and 2020). Your equation would look something like the following, where I specified each city effect explicitly:

To see why $\mu_i$ is the city effect, suppose you winnowed down your data frame to examine observations from the fifth city, which is unique with respect to many of its stable attributes (e.g., geography, land area, number of expressways, etc.). The equation for your city of interest simplifies to the following:

In this fictitious example, $\mu_5$ represents the time-constant attributes of the fifth city. For example, the geographic attributes of the fifth city represent more or less "fixed" aspects of that city. If you differenced or demeaned this equation, your city effect will be removed. Again, to see why this works, for each $i$ suppose we calculated the average of a jurisdiction's square mileage over time $t$; the average of this time-invariant variable is that time-invariant variable, so you're differencing out any fixed characteristics of that city. In fact, any variable that does not exhibit any variation over time can be safely dropped from your analysis. Your model is using the time variation within a city to identify the effect of unemployment on auto theft.

Your second specification uses summation notation, which denotes the estimation of dummy variables for each cross-sectional unit. I wouldn't say it's a more "accurate" estimation method though. Rather, I would argue it is more notationally explicit, as a variable (i.e., $d_i$) is appended to your parameter (i.e., $\lambda_i$). But remember, you wouldn't want to use the LSDV estimator in settings with large $N$. Suppose in your example you acquired data on 50,000 individuals observed over 20 years. Do you really want to compromise your computer storage by trying to estimate 49,999 fixed effects? Demeaning is a useful trick to overcome this.

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