BS2247 Introduction to Econometrics Lecture 1: Basic Mathematical Review Dr. Kai Sun Aston Business School 1 / 22
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.” 2 / 22
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. 3 / 22
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. 3 / 22
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. 3 / 22
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) 4 / 22
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) 4 / 22
Types of Data ◮ Panel A combination of cross-sectional and time series data. It carries information about how individual observations vary over time. e.g., British people’s annual wage from 2001 to 2010. This course will focus on using the cross-sectional data. 5 / 22
Types of Data ◮ Panel A combination of cross-sectional and time series data. It carries information about how individual observations vary over time. e.g., British people’s annual wage from 2001 to 2010. This course will focus on using the cross-sectional data. 5 / 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. 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. 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. 6 / 22
Example: Returns to Education A typical econometric model: wage = β 0 + β 1 educ + β 2 experience + · · · + u ◮ The estimate of β 1 , is the return to education ◮ · · · includes other factors that can affect wage. ◮ u is the error term, which includes unobserved factors, like personality 7 / 22
To estimate β 1 , 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.) 8 / 22
Summation and Average Summation operator Define a sequence of n numbers { x 1 , . . . , x n } � n i =1 x i = x 1 + x 2 + . . . + x n � n i =1 c = nc , where c is a constant � n i =1 cx i = c � n i =1 x i Example: we have a sequence of 4 numbers: { x 1 = 4 , x 2 = 12 , x 3 = 2 , x 4 = 6 } . Let c = 5. � 4 i =1 x i = 4 + 12 + 2 + 6 = 24 � 4 i =1 5 = 5 + 5 + 5 + 5 = 4 × 5 = 20 � 4 i =1 5 x i = 5 � 4 i =1 x i = 5 × 24 = 120 9 / 22
Summation and Average Summation operator Define a sequence of n numbers { x 1 , . . . , x n } � n i =1 x i = x 1 + x 2 + . . . + x n � n i =1 c = nc , where c is a constant � n i =1 cx i = c � n i =1 x i Example: we have a sequence of 4 numbers: { x 1 = 4 , x 2 = 12 , x 3 = 2 , x 4 = 6 } . Let c = 5. � 4 i =1 x i = 4 + 12 + 2 + 6 = 24 � 4 i =1 5 = 5 + 5 + 5 + 5 = 4 × 5 = 20 � 4 i =1 5 x i = 5 � 4 i =1 x i = 5 × 24 = 120 9 / 22
Define a sequence of n pair of numbers { ( x 1 , y 1 ) , . . . , ( x n , y n ) } � n i =1 ( ax i + by i ) = � n i =1 ax i + � n i =1 by i = a � n i =1 x i + b � n i =1 y i Common mistakes: � n i =1 x i � n x i i =1 y i (Example: 4 5 + 12 14 � = 4+12 y i � = 5+14 ) � n i =1 i =1 x i ) 2 (Example: 4 2 + 12 2 � = (4 + 12) 2 ) � n i =1 x 2 i � = ( � n � n � n i =1 x i y i i =1 y i i =1 x i (Example: 4 × 5+12 × 14 � = 5+14 � = 4+12 ) � n � n i =1 x 2 4 2 +12 2 i 10 / 22
Define a sequence of n pair of numbers { ( x 1 , y 1 ) , . . . , ( x n , y n ) } � n i =1 ( ax i + by i ) = � n i =1 ax i + � n i =1 by i = a � n i =1 x i + b � n i =1 y i Common mistakes: � n i =1 x i � n x i i =1 y i (Example: 4 5 + 12 14 � = 4+12 y i � = 5+14 ) � n i =1 i =1 x i ) 2 (Example: 4 2 + 12 2 � = (4 + 12) 2 ) � n i =1 x 2 i � = ( � n � n � n i =1 x i y i i =1 y i i =1 x i (Example: 4 × 5+12 × 14 � = 5+14 � = 4+12 ) � n � n i =1 x 2 4 2 +12 2 i 10 / 22
From summation to average (i.e., mean) The average of the sequence (or sample) is � n x = 1 ¯ i =1 x i n or � n i =1 x i = n ¯ x Example: � 4 x = 1 i =1 x i = 1 ¯ 4 (4 + 12 + 2 + 6) = 6 4 � 4 i =1 x i = 4¯ x = 4 × 6 = 24 11 / 22
From summation to average (i.e., mean) The average of the sequence (or sample) is � n x = 1 ¯ i =1 x i n or � n i =1 x i = n ¯ x Example: � 4 x = 1 i =1 x i = 1 ¯ 4 (4 + 12 + 2 + 6) = 6 4 � 4 i =1 x i = 4¯ x = 4 × 6 = 24 11 / 22
Deviation from the mean d i = x i − ¯ x Properties: � n i =1 d i = 0 x ) 2 = � n � n i =1 x 2 x ) 2 i =1 ( x i − ¯ i − n (¯ x ) 2 = � n � n i =1 ( x i − ¯ i =1 x i ( x i − ¯ x ) � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 x i y i − n ¯ x ¯ y � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 x i ( y i − ¯ y ) � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 y i ( x i − ¯ x ) 12 / 22
Deviation from the mean d i = x i − ¯ x Properties: � n i =1 d i = 0 x ) 2 = � n � n i =1 x 2 x ) 2 i =1 ( x i − ¯ i − n (¯ x ) 2 = � n � n i =1 ( x i − ¯ i =1 x i ( x i − ¯ x ) � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 x i y i − n ¯ x ¯ y � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 x i ( y i − ¯ y ) � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 y i ( x i − ¯ x ) 12 / 22
Deviation from the mean d i = x i − ¯ x Properties: � n i =1 d i = 0 x ) 2 = � n � n i =1 x 2 x ) 2 i =1 ( x i − ¯ i − n (¯ x ) 2 = � n � n i =1 ( x i − ¯ i =1 x i ( x i − ¯ x ) � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 x i y i − n ¯ x ¯ y � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 x i ( y i − ¯ y ) � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 y i ( x i − ¯ x ) 12 / 22
Deviation from the mean d i = x i − ¯ x Properties: � n i =1 d i = 0 x ) 2 = � n � n i =1 x 2 x ) 2 i =1 ( x i − ¯ i − n (¯ x ) 2 = � n � n i =1 ( x i − ¯ i =1 x i ( x i − ¯ x ) � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 x i y i − n ¯ x ¯ y � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 x i ( y i − ¯ y ) � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 y i ( x i − ¯ x ) 12 / 22
Deviation from the mean d i = x i − ¯ x Properties: � n i =1 d i = 0 x ) 2 = � n � n i =1 x 2 x ) 2 i =1 ( x i − ¯ i − n (¯ x ) 2 = � n � n i =1 ( x i − ¯ i =1 x i ( x i − ¯ x ) � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 x i y i − n ¯ x ¯ y � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 x i ( y i − ¯ y ) � n y ) = � n i =1 ( x i − ¯ x )( y i − ¯ i =1 y i ( x i − ¯ x ) 12 / 22
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