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IBEB Methods & Techniques Summary (Grade 10)

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With this summary for the IBEB course Methods & Techniques, you have everything you need to succeed! It includes both content from the book, as well as from lecture slides. Also, it shows how to do some of the most difficult exam questions. (FEB12012X / FEB12012)

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  • July 22, 2019
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  • 2018/2019
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Chapter 1:
Estimation of Causal effects

Randomized controlled trials: ​a method of estimating causal effects:
- Control group: does not receive treatment
- Treatment group: receives treatment
- Difference between the groups: causal effect of treatment
- Do not need to know a causal effect to make a forecast


Data: Sources and Types
Experimental data​: data from controlled experiments investigating causal effects
Observational data​: data from outside the experiment setting (surveys, historical records
etc)
- Difficult to find causal effects (as no treatment or control groups)

Cross-sectional data​: data on different entities / on many different subjects
- E.g. GDP of many different countries
- observation number: ​ arbitrarily assigned number to one subject that organizes the
data

Time series data​: data from single entity / subject collected at multiple time periods
- E.g. growth rate of GDP in US over time
- Can be used to study trends and forecast

Panel / longitudinal data​: multiple entities / subjects in which each entity is observed at two
or more time periods.
- Combination of cross-sectional and time series data



Chapter 3.5: Estimation of Causal
Effects using Experimental data
Causal effect of a treatment / treatment effect: ​expected effect on the outcome of interest
of the treatment as measured in an ideal randomized controlled experiment
- Difference of two conditional expectations
- E(Y | X = x) - E(Y | X = 0)
- EV of Treatment group ​x​ - EV of control group

Causal effect of binary controlled experiment​: difference in mean outcomes
- Causal effect = mean outcome treatment group - control group

,Ecological Fallacy​: erroneously drawing conclusions about individuals solely from the
observations of higher aggregations
- Cannot draw conclusions at individual level from aggregate analysis

Conceptualization​: The process through which we specify what we mean when we use a
particular term in research.
- Defining the meaning of words used in the study
- Typically difficult In social sciences
- Defining abstract ideas with specific characteristics..

Operationalization​: specifying how a variable or concept will be measured in a specific
study.

Operationalization​: criteria for measurement quality
1. Reliability​:
a. Quality of measurement method
b. Repeated observations of same phenomenon result in the same data
2. Validity​:
a. A valid measure accurately reflects the concept it is intended to measure
b. You actually measure what you want to measure



Chapter 4: Linear Regression
Linear regression model​: Y​i​ = β​0​ + β​1​X​i​+ u​i
- Y​i is
​ the dependent variable / regressand / left-hand variable;
- X​i​ is the independent variable / regressor / right-hand variable;
- β​0​ + β​1​X​i​ is the population regression function;
- Average​ relationship between X and Y
- β​0​ is the intercept
- Only interpretable if value of 0 for X is reasonable
- β​1​ is the slope
- How much Y​i​ changes if X​i​ changes by 1
- u​i​ is the error term
- Vertical distance from observation to regression line
- Contains all the other factors besides X that determine the value of the
dependent variable

n
1
Sample covariance​: n−1 ∑ (X i − X avr )(Y i − Y avr )
i=1
- Why n-1? → corrects for a slight downward bias introduced because two regression
coefficients were estimated
- Tells us if X and Y tend to move in the same (+) or opposite directions (-)
- Units: units of X × units of Y
- n = sample size
- X​i​ or Y​i​ = value of X or Y for observation i
- X​avr​ or Y​avr​ ​= sample average of X or Y

, s XY
Sample correlation (coefficient)​: r XY = sX sY
- s​XY​ = covariance, s​x​ = st. dev of X, s​Y​ is st. dev of Y
- Always between -1 and 1
- Strength of linear relationship between X and Y

How does OLS work​:
n
- OLS finds β​0​ and β​1​ so that ∑ (Y i − β 0 − β 1 X i ) 2 is minimized
i=1
- Vertical distance between observation Y​i​ and line is: Y i − β 0 − β 1 X 1
- Squared distances must be minimized to fit the line best
- Why squared distance?
- Accounts for both positive and negative distances
- Puts more weight on points closer to the line

n
∑ (X i −X avr )(Y i −Y avr )
i=1 s XY Cov(XY )
OLS Estimator of​ β​1​: β1= n = 2 =
∑ (X i −X) 2 sX s 2x
i=i


OLS Estimator of​ β​0​ : β 0 = Y − β 1 X

Predicted values​: Y​pred​i​ = β​0​ + β​1​X​i
Residuals​: û​i​ = Y​i​ - Y​i​pred



Measures of fit
R Squared (R​2​)​: how well the regression fits the data (1 is perfect, 0 is not at all)
- Measures the fraction of the variance of Y​i​ that is explained by X​i
- R​2​ = corr(Y​i​,X​i​)​2
- R​2​ = corr(Y​pred​i​,Y​actual​i​)​2



ESS SSR
R​2​ =​ T SS =1- T SS
n
- Total variation​ (Total Sum of Squares): T SS = ∑ (Y i − Y ) 2
i=1
- Note: ​actual​ observation Y​i
n ︿
- Explained variation​: E SS = ∑ (Y i − Y ) 2
i=1
- Note: ​predicted​ Y
n
︿
- Sum of Squared Residuals​: S SR = ∑ ui 2
i=1


Standard Error of Regression (SER)​:

, - estimator of the standard deviation of the regression error u​i
- Measure of spread of the observations around the regression line
- If SER is large → predictions often very different from actual values

2

2 SSR
S ER = s︿u = s ︿
u
where s = ︿
u n−2
- Divide by n-2 because there are two degrees of freedom (two coefficients were
estimated, namely β​0​ and β​1​)

Assumptions of Ordinary Least-Square Regression​:
1. None of the regressors is correlated with the error term
a. ‘Zero conditional mean assumption’ → E(u​i​ | X​i​) = 0
b. EV of u​i​ is always 0, regardless of X​i​ → corr(u​i​ , X​i​) = 0
c. If X​i​ is taken at random → conditions holds
2. Observations are independent and identically distributed (i.i.d.)
a. If (X​i​, Y​i​) have the same distribution (e.g. drawn from same population)
b. Are independent
c. Does NOT hold for:
i. Time series
ii. Panel data (multiple observations for the same entity)
iii. Non-representative samples
3. Large outliers are unlikely
a. OLS is sensitive to large outliers
b. 0 < E(X​i​4​) < ∞ and 0 < E(Y​i​4​) < ∞


Sampling Distributions of OLS Estimators
β​0​ and β​1​ are random variables with probability
distribution
- As they are computed from a random sample
- Different sample → different estimates
- With many large samples: est. β​1​ follows normal
distribution, centered at ​actual​ β​1

Why is β​1​ normally distributed?
- Central limit theorem​: variables in large enough samples with a finite level of variance
follow approximate normal distribution pattern
- All requirements of CLT for β​1​ are fulfilled → β​1​ follows approx. normal distribution


Mean of OLS Estimator and unbiasedness
Means of estimated β​0​ and β​1
- E(β​0​est​) = β​0 (EV of estimated β​0​ equals true β​0​)
est​
- E(β​1​ ) = β​1 (EV of estimated β​1​ equals true β​0​)
- → OLS Estimators are unbiased

Unbiasedness of β​1​: (see slides lecture 3 wk 1)

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