BS2247 Introduction to Econometrics Lecture 1: Basic Mathematical - - PowerPoint PPT Presentation

BS2247 Introduction to Econometrics Lecture 1: Basic Mathematical Review

• Dr. Kai Sun

Aston Business School

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Welcome to BS2247: Introduction to Econometrics I

What is Econometrics? Econometrics = Economics + Statistics “Econometrics is based upon the development of statistical methods for estimating economic relationships, testing economic theories, and evaluating and implementing government and business policy.”

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Why study Econometrics?

◮ It allows one to test economic theory or to estimate a

relationship with real world data

◮ The real world data are non-experimental, or observational ◮ We can use econometrics to evaluate the effectiveness of a

program. Say we are interested in knowing if a publicly funded job training program can increase people’s wage.

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Why study Econometrics?

◮ It allows one to test economic theory or to estimate a

relationship with real world data

◮ The real world data are non-experimental, or observational ◮ We can use econometrics to evaluate the effectiveness of a

program. Say we are interested in knowing if a publicly funded job training program can increase people’s wage.

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Why study Econometrics?

◮ It allows one to test economic theory or to estimate a

relationship with real world data

◮ The real world data are non-experimental, or observational ◮ We can use econometrics to evaluate the effectiveness of a

program. Say we are interested in knowing if a publicly funded job training program can increase people’s wage.

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Types of Data

◮ Cross-sectional

Each observation is an individual, firm, etc. with information at a point in time (e.g., British people’s annual wage in 2012)

◮ Time series

Each observation is for a particular time period (e.g., UK GDP from 2001 to 2010)

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Types of Data

◮ Cross-sectional

Each observation is an individual, firm, etc. with information at a point in time (e.g., British people’s annual wage in 2012)

◮ Time series

Each observation is for a particular time period (e.g., UK GDP from 2001 to 2010)

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Types of Data

◮ Panel

A combination of cross-sectional and time series data. It carries information about how individual observations vary

• ver time.

e.g., British people’s annual wage from 2001 to 2010. This course will focus on using the cross-sectional data.

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Types of Data

◮ Panel

A combination of cross-sectional and time series data. It carries information about how individual observations vary

• ver time.

e.g., British people’s annual wage from 2001 to 2010. This course will focus on using the cross-sectional data.

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Correlation vs. Causality

◮ Econometric models aim to estimate causal, or ceteris paribus,

relationships. ceteris paribus: holding everything else constant.

◮ For example, simply knowing the correlation between

education and wage is not enough. Policy makers are more interested in knowing how an increase in education causes wage to increase, ceteris paribus.

◮ When examining the impact of education on wage, we should

hold constant other factors, such as experience, gender, race, etc., that can potentially affect wage.

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Correlation vs. Causality

◮ Econometric models aim to estimate causal, or ceteris paribus,

relationships. ceteris paribus: holding everything else constant.

◮ For example, simply knowing the correlation between

education and wage is not enough. Policy makers are more interested in knowing how an increase in education causes wage to increase, ceteris paribus.

◮ When examining the impact of education on wage, we should

hold constant other factors, such as experience, gender, race, etc., that can potentially affect wage.

6 / 22

Correlation vs. Causality

◮ Econometric models aim to estimate causal, or ceteris paribus,

relationships. ceteris paribus: holding everything else constant.

◮ For example, simply knowing the correlation between

education and wage is not enough. Policy makers are more interested in knowing how an increase in education causes wage to increase, ceteris paribus.

◮ When examining the impact of education on wage, we should

hold constant other factors, such as experience, gender, race, etc., that can potentially affect wage.

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Example: Returns to Education

A typical econometric model: wage = β0 + β1educ + β2experience + · · · + u

◮ The estimate of β1, is the return to education ◮ · · · includes other factors that can affect wage. ◮ u is the error term, which includes unobserved factors, like

personality

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To estimate β1, we must have necessary statistics knowledge.

◮ The first two lectures will cover some basic statistics. ◮ For the rest of the lectures, we will refer to these basics from

time to time.

◮ Course logistics (homework, tutorial, office hours, etc.)

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Summation and Average

Summation operator Define a sequence of n numbers {x1, . . . , xn} n

i=1 xi = x1 + x2 + . . . + xn

n

i=1 c = nc, where c is a constant

n

i=1 cxi = c n i=1 xi

Example: we have a sequence of 4 numbers: {x1 = 4, x2 = 12, x3 = 2, x4 = 6}. Let c = 5. 4

i=1 xi = 4 + 12 + 2 + 6 = 24

4

i=1 5 = 5 + 5 + 5 + 5 = 4 × 5 = 20

4

i=1 5xi = 5 4 i=1 xi = 5 × 24 = 120

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Summation and Average

Summation operator Define a sequence of n numbers {x1, . . . , xn} n

i=1 xi = x1 + x2 + . . . + xn

n

i=1 c = nc, where c is a constant

n

i=1 cxi = c n i=1 xi

Example: we have a sequence of 4 numbers: {x1 = 4, x2 = 12, x3 = 2, x4 = 6}. Let c = 5. 4

i=1 xi = 4 + 12 + 2 + 6 = 24

4

i=1 5 = 5 + 5 + 5 + 5 = 4 × 5 = 20

4

i=1 5xi = 5 4 i=1 xi = 5 × 24 = 120

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Define a sequence of n pair of numbers {(x1, y1), . . . , (xn, yn)} n

i=1(axi + byi) = n i=1 axi + n i=1 byi = a n i=1 xi + b n i=1 yi

Common mistakes: n

i=1 xi yi = n

i=1 xi

n

i=1 yi (Example: 4

5 + 12 14 = 4+12 5+14)

n

i=1 x2 i = (n i=1 xi)2 (Example: 42 + 122 = (4 + 12)2) n

i=1 xiyi

n

i=1 x2 i

=

n

i=1 yi

n

i=1 xi (Example: 4×5+12×14

42+122

= 5+14

4+12)

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Define a sequence of n pair of numbers {(x1, y1), . . . , (xn, yn)} n

i=1(axi + byi) = n i=1 axi + n i=1 byi = a n i=1 xi + b n i=1 yi

Common mistakes: n

i=1 xi yi = n

i=1 xi

n

i=1 yi (Example: 4

5 + 12 14 = 4+12 5+14)

n

i=1 x2 i = (n i=1 xi)2 (Example: 42 + 122 = (4 + 12)2) n

i=1 xiyi

n

i=1 x2 i

=

n

i=1 yi

n

i=1 xi (Example: 4×5+12×14

42+122

= 5+14

4+12)

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From summation to average (i.e., mean) The average of the sequence (or sample) is ¯ x = 1

n

n

i=1 xi

• r

n

i=1 xi = n¯

x Example: ¯ x = 1

4

4

i=1 xi = 1 4(4 + 12 + 2 + 6) = 6

4

i=1 xi = 4¯

x = 4 × 6 = 24

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From summation to average (i.e., mean) The average of the sequence (or sample) is ¯ x = 1

n

n

i=1 xi

• r

n

i=1 xi = n¯

x Example: ¯ x = 1

4

4

i=1 xi = 1 4(4 + 12 + 2 + 6) = 6

4

i=1 xi = 4¯

x = 4 × 6 = 24

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Deviation from the mean

di = xi − ¯ x Properties: n

i=1 di = 0

n

i=1(xi − ¯

x)2 = n

i=1 x2 i − n(¯

x)2 n

i=1(xi − ¯

x)2 = n

i=1 xi(xi − ¯

x) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xiyi − n¯

x¯ y n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xi(yi − ¯

y) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 yi(xi − ¯

x)

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Deviation from the mean

di = xi − ¯ x Properties: n

i=1 di = 0

n

i=1(xi − ¯

x)2 = n

i=1 x2 i − n(¯

x)2 n

i=1(xi − ¯

x)2 = n

i=1 xi(xi − ¯

x) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xiyi − n¯

x¯ y n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xi(yi − ¯

y) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 yi(xi − ¯

x)

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Deviation from the mean

di = xi − ¯ x Properties: n

i=1 di = 0

n

i=1(xi − ¯

x)2 = n

i=1 x2 i − n(¯

x)2 n

i=1(xi − ¯

x)2 = n

i=1 xi(xi − ¯

x) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xiyi − n¯

x¯ y n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xi(yi − ¯

y) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 yi(xi − ¯

x)

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Deviation from the mean

di = xi − ¯ x Properties: n

i=1 di = 0

n

i=1(xi − ¯

x)2 = n

i=1 x2 i − n(¯

x)2 n

i=1(xi − ¯

x)2 = n

i=1 xi(xi − ¯

x) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xiyi − n¯

x¯ y n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xi(yi − ¯

y) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 yi(xi − ¯

x)

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Deviation from the mean

di = xi − ¯ x Properties: n

i=1 di = 0

n

i=1(xi − ¯

x)2 = n

i=1 x2 i − n(¯

x)2 n

i=1(xi − ¯

x)2 = n

i=1 xi(xi − ¯

x) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xiyi − n¯

x¯ y n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xi(yi − ¯

y) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 yi(xi − ¯

x)

12 / 22

Deviation from the mean

di = xi − ¯ x Properties: n

i=1 di = 0

n

i=1(xi − ¯

x)2 = n

i=1 x2 i − n(¯

x)2 n

i=1(xi − ¯

x)2 = n

i=1 xi(xi − ¯

x) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xiyi − n¯

x¯ y n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xi(yi − ¯

y) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 yi(xi − ¯

x)

12 / 22

Deviation from the mean

di = xi − ¯ x Properties: n

i=1 di = 0

n

i=1(xi − ¯

x)2 = n

i=1 x2 i − n(¯

x)2 n

i=1(xi − ¯

x)2 = n

i=1 xi(xi − ¯

x) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xiyi − n¯

x¯ y n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 xi(yi − ¯

y) n

i=1(xi − ¯

x)(yi − ¯ y) = n

i=1 yi(xi − ¯

x)

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Proportionate/Percentage/Percentage-point change

Proportionate change Exact: (x2 − x1)/x1 Approximate: ln x2 − ln x1 Percentage change: Proportionate change ×100 Exact: [(x2 − x1)/x1] × 100 Approximate: (ln x2 − ln x1) × 100 Percentage point change: change in the percentages

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Example Question: A person’s income increases from $30,000 per year to $36,000 per year. Calculate exact and approximate proportionate change and percentage change. Answer: Exact proportionate change = 36000−30000

30000

= 0.2 Approximate proportionate change = ln 36000 − ln 30000 = 0.18 Exact percentage change = 0.2 × 100 = 20 Approximate percentage change = 0.18 × 100 = 18

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Example Question: A person’s income increases from $30,000 per year to $36,000 per year. Calculate exact and approximate proportionate change and percentage change. Answer: Exact proportionate change = 36000−30000

30000

= 0.2 Approximate proportionate change = ln 36000 − ln 30000 = 0.18 Exact percentage change = 0.2 × 100 = 20 Approximate percentage change = 0.18 × 100 = 18

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A rule-of-thumb that economists always use

∆ ln x = 1 = ⇒ %∆x = 100 That is, that the natural logarithm of x changes by 1 unit implies that x changes by 100%. To understand this, think like this: If proportionate change is 1, then percentage change is 100.

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Application of the rule-of-thumb

Model 1 (log-log): ln y = β0 + β1 ln x + u Interpret β1:

• when ln x increases by 1 unit, ln y increases by β1 units =

⇒

• when x increases by 100%, y increases by β1 × 100% =

⇒

• when x increases by 1%, y increases by β1%

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Application of the rule-of-thumb

Model 2 (log-level): ln y = β0 + β1x + u Interpret β1:

• when x increases by 1 unit, ln y increases by β1 units =

⇒

• when x increases by 1 unit, y increases by β1 × 100%

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Application of the rule-of-thumb

Model 3 (level-log): y = β0 + β1 ln x + u Interpret β1:

• when ln x increases by 1 unit, y increases by β1 units =

⇒

• when x increases by 100%, y increases by β1 units =

⇒

• when x increases by 1%, y increases by β1/100 units

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Functions in econometric models

• 1. Linear functions

(a) y = β0 + β1x + u ∆y = β1∆x = ⇒ β1 = ∆y

∆x is the marginal effect of x on y.

(b) y = β0 + β1x1 + β2x2 + u ∆y = β1∆x1 + β2∆x2 = ⇒ β1 = ∆y

∆x1 if ∆x2 = 0

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Functions in econometric models

• 2. Quadratic functions

y = β0 + β1x + β2x2 + u Important properties: (a) U-shaped if β2 > 0 (b) Inverse U-shaped if β2 < 0 Need a little bit calculus to calculate the marginal effect of x:

∆y ∆x = β1 + 2β2x

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Functions in econometric models

• 3. Logarithm functions

y = ln x + u (x > 0) Properties of natural logarithm: (a) ln(x1x2) = ln x1 + ln x2 (b) ln x1

x2 = ln x1 − ln x2

(c) ln xc = c ln x

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Reading

Appendix A, Introductory Econometrics - A Modern Approach, 4th Edition, J. Wooldridge

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