Two-Period Panel Data Analysis

According to CANA, more people are choosing cremation because it is (1) affordable, (2) environmentally friendly, (3) easier given our “geography and population mobility, ” and (4) increasingly acceptable to religious groups.

http://www.kates-boylston.com/NewsPage.aspx?newsID=2122

Cremation rate (%)

Natives, born in state (%)Year dummy

The 35 states are: AL, AZ, AR, CO, CT, FL, GA, ID, IN, IA, KS, KY, ME, MD, MA, MI, MN, MO, MT, NE, NV, NJ, NM, NC, ND, OR, PA, SC, SD, TX, UT, VT, WA, WI, and WY

𝐶𝑅𝑖𝑡=𝛽0+𝛿0𝑌𝑟 2000+𝛽1𝑁 𝑖𝑡+𝜀𝑖𝑡

A simple empirical specification that focuses on Boylston’s third explanation for the increasing proportion of people choosing cremation is:

(1)

Pooled OLS

𝐶𝑅𝑖𝑡=64.4+9.34 ∙𝑌𝑟 2000−0.757 ∙𝑁 𝑖𝑡

0 10 20 30 40 50 60 70 80 90 100051015202530354045505560657075

Cremation

Rate (%)

Native (% born in state)

𝑠𝑙𝑜𝑝𝑒=−0.757𝐶𝑅

𝑖𝑡=2000𝐶𝑅

𝑖𝑡=1990

9.34

0 10 20 30 40 50 60 70 80 90 100051015202530354045505560657075

Cremation

Rate (%)

Native (% born in state)

Colorado

Georgia

Fewer people living in Colorado were born there than in Georgia and a lot of the variation in Native used to estimate is coming from between states and some of the variation is coming from within states over time.

State fixed effect () captures (time-invariant and unobserved) prices, regulations, environmental attitudes, religious attitudes. If they are observable, you are better off putting them into the equation as explanatory variables.

𝜀𝑖𝑡=𝑎𝑖+𝑢𝑖𝑡

Time varying error (idiosyncratic error) —unobserved factors that affect cremation rates and vary over time

𝐶𝑅𝑖𝑡=𝛽0+𝛿0𝑌𝑟 2000+𝛽1𝑁 𝑖𝑡+𝜀𝑖𝑡

𝐶𝑅𝑖𝑡=𝛽0+𝛿0𝑌𝑟 2000+𝛽1𝑁 𝑖𝑡+𝑎𝑖+𝑢𝑖𝑡

Fixed Effects Model

(2)

(1)

Pooled OLS

Pooled OLS is not substantially different from single-time-period OLS. If you have an omitted variable problem due to stuff in the error term, pooling the data doesn’t eliminate it.

𝐶𝑅𝑖𝑡=𝛽0+𝛿0𝑌𝑟 07+𝛽1 𝑁 𝑖𝑡+𝜀𝑖𝑡

For simplicity, suppose

𝐶𝑅𝑖𝑡=𝛽0+𝛿0𝑌𝑟 07+𝛽1 𝑁 𝑖𝑡+𝛽2 𝑅2𝐸𝑖𝑡+𝑢𝑖𝑡

E⃝� – ⃝� – ⃝�+

downward bias

𝐶𝑅𝑖2000=𝛽0+𝛿0+𝛽1 𝑁 𝑖2000+𝑎𝑖+𝑢𝑖 2000

𝐶𝑅𝑖1990=𝛽0+𝛽1 𝑁 𝑖1990+𝑎𝑖+𝑢𝑖1990

∆𝐶𝑅𝑖=𝛿0+𝛽1∆𝑁 𝑖+∆𝑢𝑖

First-difference equation: eliminates

First Differences

(3)

Estimating Fixed Effects Models

𝐶𝑅𝑖𝑡=𝛽0+𝛿0𝑌𝑟 2000+𝛽1𝑁 𝑖𝑡+𝑎𝑖+𝑢𝑖𝑡(2)

Estimating First-Differencing Models

∆𝐶𝑅𝑖=𝛿0+𝛽1∆𝑁 𝑖+∆𝑢𝑖(3)

Estimating First-Differencing Models ∆𝐶𝑅𝑖=𝛿0+𝛽1∆𝑁 𝑖+∆𝑢𝑖(3)

Estimating Fixed Effects Models

𝐶𝑅𝑖𝑡=𝛽0+𝛿0𝑌𝑟 2000+𝛽1𝑁 𝑖𝑡+𝑎𝑖+𝑢𝑖𝑡(2)

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Demonstrates that models using fixed effects are using variation within states (or cities, counties, colleges, etc.) to estimate parameters

𝐶𝑅𝑖𝑡=𝛽0+𝛿0𝑌𝑟 07+𝛽1 𝑁 𝑖𝑡+𝜀𝑖𝑡

𝜀𝑖𝑡=𝑎𝑖+𝑢𝑖𝑡

∆𝐶𝑅𝑖=𝛿0+𝛽1∆𝑁 𝑖+∆𝑢𝑖

Key Assumption is uncorrelated with

This assumption holds if the idiosyncratic error (u) at each time period is uncorrelated with the explanatory variable in both time periods.

Costs and Benefits of Fixed Effects Model

Benefit—controls for unobserved factors that vary across states, cities, colleges… Costs1. More expensive data collection2. Can reduce or eliminate variation in explanatory variables.