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Worksheet: Regression Interpretation questions (1)

  1. For the following questions, refer to the following equation and its respective graph
      2. Y=α0+α1Treatment+ϵY=\alpha_0+\alpha_1Treatment+\epsilon
      image
    1. What’s the value of α^0 and α^1\hat\alpha_0 \ and\ \hat\alpha_1?
      2. ‣
      Answer
      ✅
      α^0=0  and  α^1=2\hat\alpha_0=0\ \ and\ \ \hat\alpha_1=2
      image
    2. What’s the value of α^0 and α^1\hat\alpha_0 \ and\ \hat\alpha_1?
      3. ‣
      Answer
      ✅
      α^0=4 and α^1=−10\hat\alpha_0=4 \ and\ \hat\alpha_1=-10

The following are questions that can help you hone in your understanding of a regression or your understanding of how to interpret regression.

When do countries tax wealth? Taxes are a big deal. they affect how people allocate their time, how much money government has, etc. Inheritance taxes are a particularly interesting tax policy because of the clear potential for conflict between rich and poor. Scheve and Stasavage (2012) investigated the sources of inheritance taxes by looking at tax policy and other characteristics of 19 countries for which data is available from 1816 to 2000. The data is measured every five years. Specifically the researchers looked at the relationship between inheritance taxes and who was allowed to vote. To assess if expanded suffrage led to increases or decreases in inheritance taxes, we can begin with the following model:

Inheritance taxit=β0+β1Expanded Suffragei,t−1+ϵitInheritance\ tax_{it}=\beta_0+\beta_1Expanded\ Suffrage_{i,t-1}+\epsilon_{it}

The dependent variable is the top inheritance tax rate, which is measure as a percent, and the independent variable is a dummy variable for whether all men were eligible to cote in at least half of the previous years.

  1. What does β0\beta_0 represents?
    2. ‣
    Answer
    ✅
    The mean tax rate across 19 countries and 184 years, when suffrage wasn’t expanded.
  2. What does β1\beta_1 represent?
    3. ‣
    Answer
    ✅
    The additional tax or the marginal tax change that happens on average when countries expand suffrage. Notice that is not an “increase” or a “decrease” yet since we don’t know what the number is.
  3. What’s the average difference in inheritance tax between expanded and not expanded suffrage?
    4. ‣
    Answer
    ✅
    β1\beta_1
  4. Let’s say you get:
      5. Inheritance taxit=4.75+19.33Expanded Suffragei,t−1Inheritance\ tax_{it}=4.75+19.33Expanded\ Suffrage_{i,t-1}
    1. What’s the average inheritance tax for countries without expanded suffrage? Recall that units of inheritance tax are on percent from 0-100%
      2. ‣
      Answer
      ✅
      4.75 percent.
    2. What’s the average inheritance tax for countries with expanded suffrage?
      3. ‣
      Answer
      ✅
      The average inheritance tax rate for countries that expanded suffrage is 24.07 percent. This is because we add both coefficients.
    3. What’s the average difference in inheritance tax between expanded and not expanded suffrage?
      4. ‣
      Answer
      ✅
      The average difference is 19.33 percentage points. Notice that there difference represents change of percent, which are percentage points.
  5. For the following questions refer to the following table. The outcome is inheritance tax rate.
    6. image

  1. What’s the marginal effect of having universal suffrage on inheritance tax rate in column (d)?
    2. ‣
    Answer
    ✅
    0.69 percentage point
  2. Write column (d) as an equation with numbers.
    3. ‣
    Answer
    ✅
    InheritanceTax=−521.63+0.69UniversalCoverage+0.28Year+11.76War+2.19Europe+18.71Asia+7.84NorthAmericaInheritanceTax=-521.63+0.69UniversalCoverage+0.28Year+11.76War+2.19Europe+18.71Asia+7.84NorthAmerica
  1. In an effort to better understand the effects of “Get-out-the-vote” messages on voter turnout, Gerber and Green (2005) conducted an RCT involving approximately 30,000 individuals in New Haven, CT, in 1998. One of the treatments was randomly assigned in person visits in which a volunteer visited the person's home and encouraged him or her to vote. Table 3 reflects the findings from the RCT.
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  1. What’s the marginal effect of being assign to in person contact on voting?
    2. ‣
    Answer
    ✅
    0.47-0.45=0.02, it’s 2 percentage points. (Y is a binary variable in 0/1)
  2. Fill in the values of the following tables using the values from the table above
    3. RegressionsActually ContactedVoted in 1998Assigned to in-person contactβ1α1Constantβ0α0\begin{array}{c:c:c} \colorbox{yellow}{Regressions} & \text{Actually Contacted} & \text{Voted in 1998} \\ \hline \\ \text{Assigned to in-person contact} & \beta_1 & \alpha_1 \\ & \\ Constant & \beta_0 & \alpha_0 \end {array}
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    Answer
    ✅
    β1=0.25, β0=0.03, α1=0.02, α0=0.45\beta_1=0.25,\ \beta_0=0.03,\ \alpha_1=0.02,\ \alpha_0=0.45

8. Use the figure below to answer: What’s the sign of β2\beta_2 in the following equation? Y=β0+β1X+β2X2Y=\beta_0+\beta_1X+\beta_2X^2

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Answer
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It’s negative. This is because since there is a local max, this means that the second derivative must be negative, because the rate of change of the first derivative is decreasing. The first derivative represents the slope at any point, so in order to see that the slope is decreasing, pick a point of x (say X=25) and see what the slope is, in this case it will be a positive number. Now pick a point that’s higher than your original point (say x=75), and now see the slope of this point, notice that now the slope is negative. Therefore the slope went from a positive number to a negative number, so the slope is decreasing, which means the first-derivative is decreasing, which means the second derivative is negative, and since the second derivative is the sign of β2\beta_2 we then know that β2\beta_2 is negative.

9. Use the following figure to answer the following question. The main dependent variable on all the models is wages measured in hourly dollars, while height is measured in inches. The first column’s dependent variable is wages, second is wages, third is log(wages) and fourth is log (wages).

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  1. Using column (1) What’s the relationship between adolescent height and wages?
    2. ‣
    Answer
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    One inch increase in adolescent height is associated with a $0.412 increase in hourly wages
  2. Using column (3) What’s the relationship between adolescent height and wages?
    3. ‣
    Answer
    ✅
    A one-inch increase in height is associated with 3,3 percent increase in wages
  3. Using column (1): Patricia grew about 5 inches between birthdays, how much is Patricia expected to earn now relative to her last birthday?
    4. ‣
    Answer
    ✅
    Patricia grew 5 inches, this means that Patricia will now earn 5×0.412=2.065\times0.412=2.06, or about two more dollars per hour.
  4. What’s the interpretation of the coefficient 29.316 in column (2)
    5. ‣
    Answer
    ✅
    A one percent increase in adolescent height is associated with a $0.293 increase in hourly wages
  5. What’s the interpretation of the coefficient 2.362 in column (4)
    6. ‣
    Answer
    ✅
    A one percent increase in adolescent height increases wages by 2.362 percent.
  6. I want to know how much wages increase in percent, when height increases by 15%, Which of these models would give me the quickest way to figure this out? and what’s the answer?
    7. ‣
    Answer
    ✅
    Since we want increase in wages in percent, we need one of the models where Y is logged. Since we have a change in X in percent, we want a model where X is also logged. Model 4 will give us this since they are both on logs. An increase of 15% in height is associated with a 15×2.362=35.4315\times2.362=35.43, or about 35%.
  7. I want to know how much wage increase in dollars when height increases by 20%, which of these models would give me the quickest way to figure this out? and what’s the answer?
    8. ‣
    Answer
    ✅
    In this case, we want the Y to be in hourly wages, and X to be logged. Therefore we use the second model. An increase in 20% in adolescent height means an increase in wages of 20×29.316=586.32,20\times29.316=586.32, which means an increase of $5.86 in hourly wages.
  1. Energy Efficiency promises a double whammy of benefits: if we reduce the amount of energy we can both save the world and save money. What’s not to love? In this exercise we’ll dig into how to explore this relationship. The technology innovation is a programmable thermostat, which is a device that allows the user to preset temperatures at energy-efficient levels. Another important variable is HDD “heating degree-days”, which is a measure of how cold it was in the month (it is the number of degrees that a day’s average temperature is below 65 degree Fahrenheit). Usually the relationship between HDD and temperature (measure as Therms) is positive (the colder it gets, the higher the temperature people set their thermostat). We have data of houses that use thermostat and houses that don’t. The results from an OLS analysis are below, the main outcome variable for all of these regressions is “Therms”. The cost of a therm is $1.59 per therm. The cost of the thermostat is $60
      2. image
    1. Using Model (a), What’s the main conclusion?
      2. ‣
      Answer
      ✅
      That when houses use programmable thermostat, this houses have about 13.02 fewer therms of energy. The overall conclusion is that thermostat are making people more energy efficient.
    2. Does you main conclusion change when accounting for HDD?
      3. ‣
      Answer
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      It only gets stronger! Now the difference are larger.
    3. Using results from model (b), how much money are houses who use thermostat saving? According to this model is the thermostat worth it?
      4. ‣
      Answer
      ✅
      We can start creating meaningful statements on how much $ one is saving. We just have to convert the coefficient -20.05 to $. Since cost of a therm is $1.59, then this means savings of about $31.87 per month. Since the cost of the thermostat is $60, one could easily pay for this in the span of two month.
    4. Does it make sense that the programmable thermostat should save $30 in the middle of the summer? This indicates that the cost-savings depend on the weather outside. It makes more sense to think about the effects of the thermostat with respect to temperature outside. Therefore we focus on model (c). What’s the interpretation of the number -0.48?
      5. ‣
      Answer
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      This is the difference in therms for houses that have vs. don’t have thermostat when HDD is 0 (meaning the weather is warm for the whole month). This coefficient seems small relative to the others, and it is not statistically significant. Notice that this is consistent with the idea that programmable thermostat should not reduce heating costs when the furnace isn’t running.
    5. Using Model (c), what is the effect of the thermostat when HDD is 500?
      6. ‣
      Answer
      ✅
      The effect of the thermostat is β^1+β^3×500\hat\beta_1+\hat\beta_3\times500=−0.48−0.062×500=−31.48-0.48-0.062\times500=-31.48, this means that the thermostat help reduce therms by 31.48, lowering the bill by $50.05
    6. Using Model (c) What’s the average therm use for houses that don’t have a thermostat is particular hot months?
      7. ‣
      Answer
      ✅
      We can think of this as HDD=0, and so the answer will be 4.24

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