Ceteris Paribus

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Ceteris Paribus

• We had economic growth of 2, unemployment was
  low at 5, inflation was under control (2) and
  there were no strikes in the previous year
• What governments VOTE share would we predict?
• Growtht2, Unempt5, Infltiont2, Strikest-10

• What if the government does some gov expenditures
  and the economy grows by 3 rather than 2? Other
  things being equal (ceteris paribus)

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Role of the Intercept in Regressions

• The intercept is the y value that we would
  predict if all explanatory variables take a value
  equal to 0
• Example
• yab1x1b2x2b3x3b4x4e with x1x2x3x40
• yab10b20b30b40a

y
yab1x1
ya
x10
x1
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Role of the Intercept in Regressions

• If the range of x values does not include 0, the
  intercept should not be interpreted as if x10

y
ya
yb1x1
x1
x10

• The intercept is nevertheless crucial
• Without the intercept, we assume a0, i.e. we
  would force the regression line to go through the
  origin (x, y)(0, 0)
• ? Consequences on the estimated slope coefficient
  b

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QM1 Week 7Dummy Variables

• Dr Alexander Moradi
• University of Oxford, Dept. of Economics
  GPRG/CSAE
• Email alexander.moradi_at_economics.ox.ac.uk

5
9. Dummy Variables

• Qualitative variables do not have an ordinal
  scale
• Dummy variables make it possible to incorporate
  qualitative factors into regression models
• A dummy variable has only 2 values ? 0, 1
• We have to define which event is assigned the
  value one and which is assigned the value zero
• Examples
• FEMALE1, if female FEMALE0, if male
• MALE1, if male MALE0, if female
• HISTORIAN1, if historian HISTORIAN0 otherwise
• UK1, if UK UK0 if other country
• SKILLED1, if skilled worker SKILLED0
  otherwise
• Dummy variables are also called binary variable
  or zero-one variable

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9.1 Dummy Variables

• Relationship between age and income in South India

Wage differential
Wages increase with the age of employees
Independent of age female employees earn less
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9.1 Incorporating Qualitative Factors

• 1. Running two separate regressions, one
  regression for each gender
• WAGEMaMbMAGEMeM for men
• WAGEFaFbFAGEFeF for women
• Advantages
• Makes differences in the coefficients visible
• By using confidence intervals we can test for
  significant differences, e.g. is aM and bM
  significantly different from aF and bF
  respectively
• Disadvantages
• We need to look at a and b jointly small
  differences in the slope coefficient b can create
  large differences in the intercept a
• Reduced number of observations in two separate
  regressions ? higher SE ? lower t-values ? lower
  precision of regression coefficients

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9.1 Using a Dummy Variable

• 2. Alternative Only the intercept is different
  (aM?aF), the slope coefficients are identical for
  both groups (bM?bF)
• Dummy variable
• FEMALE1, if female
• FEMALE0, if male
• Regression model WAGEab1FEMALEb2AGEe
• Interpretation
• Wage of a female employee at age 30
• WAGEab11b230ab1b230
• Wage of a male employee at age 30
• WAGEab10b230a b230
• b2 Effect of an increase in age by 1 year
• b1 Wage differential or what women earn holding
  age like a man

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9.1 Using a Dummy Variable

• The dummy variable is like a change in the
  intercept (for female employees), parallel upward
  (b1gt0)/downward (b1lt0) shift of the regression
  line/plane
• Advantages
• Larger sample size The wage-age pattern of both
  men and women gives us more confidence
• t-value for the coefficient of the dummy variable
  b1 indicates significance of wage differential.
  If significantly negative Holding other
  characteristics like age constant, female
  employees earn less

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9.2 Using Dummy Variables for Multiple Categories

• Qualitative characteristics are not limited to
  two dimensions
• Multiple categories More than two categories
• Example
• Society Lower / Middle/ Upper class
• Industries Manufacturing/ Textiles/ Food
  processing/ Chemicals etc.
• If we use one variable and code it with certain
  values, we would assume an order that is not
  necessarily true
• Example
• Variable CLASS with Lower class1, Middle
  class2, Upper class3
• Middle class members (CLASS2) are twice as good
  as lower class members (CLASS1). Upper class
  members (CLASS3) are thrice as good as lower
  class members
• If we have g groups or categories, we need to
  include g-1 dummy variables (including g groups
  would result in the dummy variable trap)

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9.3 Example

• Regression with two dummy variables
• LOWCLASS1, if lower class, 0 otherwise
• MIDCLASS1, if middle class, 0 otherwise

• What is the income of a member of the lower and
  middle class respectively?

• Regression with an ordinal variable CLASS1, 2, 3

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9.3 Reference Category

• The intercept reflects the predicted outcome of y
  if all dummy variables are equal to 0
• ? Intercept picks up the category for which no
  dummy variable was included. This category is the
  reference category
• The coefficients of the dummy variables express
  the effect compared to the reference category
• ? The interpretation does not change when we
  choose a different reference category
• ? Regression coefficients of the dummy variables
  and intercept adjust accordingly

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9.4 Interaction Terms

• In order to test the effect of a combination of
  characteristics, we take the product of the two
  dummy variables
• ? creates a new dummy variable
• Value of the new dummy variable1, if value was 1
  in both of the original dummy variables
• Value of the new dummy variable0, if value was 0
  in at least one of the original dummy variables
• ? Regression coefficient indicates the effect
  that the combination of characteristics has (as
  opposed to the isolated effect that is given by
  the coefficients of each of the two original
  dummy variables)

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9 Exercise Dummy Variables

• Dataset india.dta
• Data on income and background characteristics of
  261 employees in a South Indian city
• Estimate the model Ln(WI)ab1AGEb2
  EDUb2FEMALEe. Interpret the results
• Generate an interaction term FEMALE_EDU1 if
  female and secondary, FEMALE_EDU0 otherwise
• Add the interaction term to the model in (1).
  Interpret the regression coefficient of
  interaction term
• What is the expected income of a unskilled,
  female employee at age 30 years?
• What is the expected income of a skilled, male
  employee at age 30 years?
• What is the expected income of a skilled, female
  employee at age 30 years?
• Data set weimar_election.dta
• Run a regression of Nazis percentage of votes on
  the unemployment rate, share of workers,
  Catholics, and farmers. Add dummy variables for
  the time of the general election. Interpret the
  results
• What could have caused unemployment to become
  insignificant? Hint Calculate the mean
  unemployment and NAZI votes for each of the four
  elections
• Is the model specification with dummy variables
  appropriate?

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9 STATA commands
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9 Homework Exercises Week 7

• Read chapter 10.1 of Feinstein Thomas (p.
  280-291)
• Do the following exercises from Feinstein
  Thomas (p. 295-299) 2, 4
• Dataset 1699_RELIEF.DTA
• Commands to be used in 2
• gen sussex1 if county2
• replace sussex0 if county!2
• Hint You find Boyers regression model on p.
  472
• regress relief cottind allotmnt london farmers
  wealth density childall subsidy grain workhse
  roundsmn labrate sussex
• regress relief cottind allotmnt london farmers
  wealth density childall subsidy grain workhse
  roundsmn labrate

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9 Homework Exercises Week 7

• Commands to be used in 4
• generate wageincome/2.6
• (Alternatively, you can try the menu under Data/
  Create or change variables/ create new variable)
• generate graind1 if graingt20
• replace graind0 if grainlt20
• Hint You can follow this procedure for coding
  the LON dummy variables. Alternatively, you can
• srecode LONlondon, min(0) max (100) step(25)
• (Hint srecode was introduced in Week 1)
• replace LON100 if londongt100
• Hint To get an idea about the coding of new
  london_cat variable, try
• tab london LON
• xi i.LON, noomit
• Hint Use the data browser to take a look at the
  newly generated dummy variables
• regress wage _ILON_0 _ILON_25 _ILON_50 _ILON_75
  graind

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8 Homework Exercises Week 7

• The effect of including a dummy variable for a
  single observation is identical to excluding this
  observation from the regression. Explain!
• Use the dataset Depression.dta
• Estimate a multiple linear regression. Use the
  exchange rate (EXCHANGE), the real wage
  (REALWAGE) and the discount rate (INTEREST) to
  explain industrial production in 1935 (1929100).
  Interpret your results
• Hint reg prod exchange realwage interest if
  year1935
• Exclude insignificant variables from the
  regression model
• Compare your results with those obtained from
  simple linear regressions Report the results in
  one regression table with a column for each
  regression and interpret the differences
• The regression coefficient of the real wage
  differs in 4b) and 4c). Is multicollinearity the
  reason for this? Explore the correlations between
  the explanatory variables. What would you
  conclude? Does REALWAGE possibly pick up the
  effect of EXCHANGE?
• Hint check for correlations between the
  independent variables using the corr command. Do
  not forget to restrict the sample to year1935

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8 Homework Exercises Week 7

• Run the following STATA commands. Explain what
  the commands do Hint Use STATAs Help/Stata
  Command
• tsset country_id year
• generate dip((prod-L1.prod)/L1.prod)100
• label variable dip Annual Growth of Industrial
  Production (in )
• generate inflation((prices-L1.prices)/L1.prices)
  100
• replace drealwage ((realwage-L1.realwage)/L1.real
  wage)100
• generate dexchange((exchange-L1.exchange)/L1.exch
  ange)100
• regress dip inflation dexchange drealwage
• Interpret the regression results in 3e) Step 7.
  Is the specification of variable DEXCHANGE
  appropriate to model the effect of devaluation?
  Can the effect of inflation be taken as
  causality?
• How rigid were nominal wages? Test whether
  nominal wages adjusted to inflation, i.e. nominal
  wages decrease with deflation. Interpret the
  results
• generate dwage ((wage-L1.wage)/L1.wage)100
• generate inflation_year_priorL1.inflation
• regress dwage inflation_year_prior