Fixed Effects and Random Effects

One of the major benefits from using panel data as compared to cross-section data on individuals is that it enables us to control for individual heterogeneity. Not controlling for these unobserved individual specific effects leads to bias in the resulting estimates. Consider the panel data regression 1<math display='block'> <mrow> <msub> <mi>y</mi> <mrow> <mi>i</mi><mi>t</mi> </mrow> </msub> <mo>=</mo><mi>&#x03B1;</mi><mo>+</mo><msub> <msup> <mi>X</mi> <mo>&#x2032;</mo> </msup> <mrow> <mi>i</mi><mi>t</mi> </mrow> </msub> <mi>&#x03B2;</mi><mo>+</mo><msub> <mi>u</mi> <mrow> <mi>i</mi><mi>t</mi> </mrow> </msub> <mtext>&#x2003;</mtext><mtext>&#x2009;</mtext><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><mi>N</mi><mo>;</mo><mtext>&#x2003;</mtext><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><mi>T</mi> </mrow> </math>$${{y}_{{it}}}=\alpha +{{{X}^{\prime}}_{{it}}}\beta +{{u}_{{it}}}\quad \;i=1,\ldots ,N;\quad t=1,\ldots ,T$$ with i denoting individuals and t denoting time. The panel data is balanced in that none of the observations is missing whether randomly or non-randomly due to attrition or sample selection. α is a scalar, β is K Õ 1 and X _it is the itth observation on K explanatory variables. Most panel data applications utilize a one-way error component model for the disturbances, with 2<math display='block'> <mrow> <msub> <mi>u</mi> <mrow> <mi>i</mi><mi>t</mi> </mrow> </msub> <mo>=</mo><msub> <mi>&#x03BC;</mi> <mi>i</mi> </msub> <mo>+</mo><msub> <mi>&#x03BD;</mi> <mrow> <mi>i</mi><mi>t</mi> </mrow> </msub> </mrow> </math>$${{u}_{{it}}}={{\mu }_{i}}+{{\nu }_{{it}}}$$ where μ _i denotes the unobservable individual specific effect and v _it denotes the remainder disturbance. For example, in an earnings equation in labour economics, y _it will measure earnings of the head of the household, whereas X _it may contain a set of variables like experience, education, union membership, sex, or race. Note that μ _i is time-invariant and it accounts for any individual specific effect that is not included in the regression. In this case we could think of it as the individual’s unobserved ability. The remainder disturbance v_it varies with individuals and time and can be thought of as the usual disturbance in the regression. If the μ _i ’s are assumed to be fixed parameters to be estimated, we get the fixed effects (FE) model.