Fundementals to Mathematics of Econometrics

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Lecture slides to accompany mathematic principles of undergraduate econometrics course

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    Jing-Woei Chien

    Chapters 1,2,3 and 4

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    Chapter 1

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    Nature of Mathematical EconomicsApproach to economic analysis.

    Econometrics is typically concerned with the

    measurement of economic data. Use of mathematical theorems and statements to

    express findings and conclusions.

    Application of maths to purely theoretical aspects of

    economic analysis with little concern about problemsinvolving statistics.

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    Chapter 2

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    Economic Models

    Theoretical framework. Not necessarily mathematical.

    Relates the number of variables to one another incertain ways.

    A mathematical model consists of a set of equationsdesigned to describe the structure of a model.

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    Mathematical ModelsVariables

    Endogenous Variables whose solution we seek in ourmodel. E.g. Prices, quantities.

    Exogenous Variables that are taken as given.Determined by external sources.

    Note that status of variables are determined by thecontext of the model.

    Constants Coefficients Joined with variables.

    Parameters Constant that can take on many differentvalues (Parametric Constant).

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    Equations and Identities Describes variables that can be related with each other.

    We typically deal with three kinds of equations:

    Definitional Sets up an identity between two alternateexpressions that have exactly the same identity. =R-C

    Behavioural Specifies the manner in which a variablebehaves in response to changes in other variables. E.g. C=2+Q

    Conditional States the requirement that needs to satisfiedin order to compute a relation. E.g. Equilibrium Qs=Qd orMR=MC

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    Real Number System Integers

    Whole Numbers either positive or negative

    Fractions

    Ratio of integers. Rational Numbers

    Any number that can be expressed as a ratio of 2 integers. Terminating or repeating decimal. E.g. 0.333333333.

    Irrational Numbers The opposite of rational numbers. E.g. Square root of 2.

    Complex (We dont deal with these) Do not exist in the real number system. E.g. square root of

    negative numbers.

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    Sets Collection of distinct objects.

    Objects in a set are elements.

    Notation: Enumeration: S={5,6}.

    Description: I={x|x is positive}.

    Finite Set: Elements are finite

    Infinite Set: Infinite elements

    Countable Set: Elements can be counted

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    Venn Diagrams Diagrammatic representations of sets:

    Area encompassed by black box defines the universal set. Allelements not contained in either A or B still fall in the box but aresaid to mutually exclusive to sets A and B.

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    Important SetsN = {0,1,2,3,}, the set of natural numbers, non negativeintegers, (occasionally IN)

    Z= { , -2, -1, 0, 1, 2,3, ), the set of integers

    Z+= {1,2,3,} set of positive integers

    Q= {p/q | p Z, q Z, and q0}, set of rational numbers

    R, the set of real numbers

    Note: Set notation is not order sensitive. It does not matter inwhich order you write your numbers. However, conventiontypically favours arranging in ascending order.

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    Note on Complements

    A = UA, where U is the universal set.A set fixed within the framework of a theory and consisting

    of all objects considered in the theory.

    If A = {x : x is bored}, then

    A = {x : x is not bored}

    AU = U

    AndU =

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    General Notational Info

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    More on Sets x S means x is an element of set S. x S means x is not an element of set S.

    A B means A is a subset of B.

    A

    B

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    More comprehensive analysis

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    Examples:

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    Set Operations Commutativity:

    Associativity:

    Distributivity:

    B U A

    B A

    (A U B) U C =

    (A B) C =

    A U (B U C)

    A (B C)

    A U (B C) =A (B U C) =

    (A U B) (A U C)(A B) U (A C)

    A B =

    A U B =

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    Some useful extensions Identity:

    Domination:

    Idempotent:

    A U = A

    A U = A

    A U = U

    A =

    A A = A

    A A = A

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    Ordered Pairs Pair of numbers whose order in a set matters.

    Most commonly applied to functions where one plotspoints on the Cartesian coordinate plane.

    The Cartesian coordinate plane is an infinite set ofpoints each of which is an ordered pair with a firstvalue x and second value y.

    Associates a y value with an x value.

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    Functions If a relation (ordered pair) is such that for each x value

    there is a unique y value, then relation described is afunction. However, the converse is not necessarilytrue! (Think of a circle and semi-circle).

    Easy way to think of it is the ruler test/vertical linetest.

    Functions are also called mapping/transformationsand hence: y=f(x) where y is the value and x is theargument.

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    Exercises:

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    Functions Contd. The y value is the image of the x value under f(x).

    Set over which x is defined is the DOMAIN of f(x).

    Set of images(y) of x under f is the RANGE of f(x). Relating to econometrics, x is referred to the

    independent variable whilst y is referred to as thedependent variable.

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    Types of Functions Constant Functions

    A function whose range consists of only one element.

    Polynomial Functions Consists of many terms in which each term contains a

    coefficient as well as a nonnegative-integer.

    Rational Functions Function in which y is expressed as a ratio of two

    polynomials in the variable x. Nonalgebraic Functions (Transcendental Functions)

    Any function expressed in terms of polynomials and/orroots of polynomials.

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    Examples:

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    Of Particular Interest

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    Exponents Indicators of the power to which a variable (or

    number) is to be raised.

    Often used as part of a function and has a number ofproperties that are useful to keep in mind at all times.

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    Exponent Rules

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    A Final Note on Functions Functions can have more than 2 independent

    variables. E.g. z=g(x,y).

    A typical example is the Cobb-Douglas productionfunction: Y= F(K,L)

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    Chapter 3

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    Equilibrium Given a set of equations, the equilibrium is obtained

    when we solve for the endogenous variables that areequivalent to the function of exogenous variables and

    parameters.

    If exogenous variables/ parameters change we willobserve a new equilibrium which leads us to classifyequilibrium as static analysis.

    In Economics, people frequently analyse partialequilibrium in an isolated market.

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    Market Model Given Qd, Qs and P.

    Assume Qd=a-bP and Qs=-c+dP where a,b,c,d >0

    We know from Economics that equilibrium is definedat the point where Qd=Qs.

    Therefore, we proceed to elliminate Qs and Qd byrenaming a single variable Q in equilibrium.

    The system of equations then yield the followingexpression: Q=a-bP=-c+dP in equilibrium.

    However, not all calculations and equations can bemanipulated so easily.

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    2-Commodity Case Given:

    10 2+

    2 + 3

    15 + 1 + 2

    Now we have to solve this system of equations to getequilibrium price and quantities in equilibrium.

    Make use of simultaneous equations.

    From this, we see thatthe quantity demanded

    of each good isdependent on the pricesof both goods. Realistic?

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    Non-linear Models Economics, however, is not restricted to linear

    relationships.

    For example: Given

    4 4 1

    We again assume Qd=Qs in equilibrium to yield:

    4 4 1which, when simplified, results to+ 4 5 0 How do we solve this?

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    The Quadratic Formula

    This is a useful formula to remember/memorise, however,this will be provided on your test/exam formula sheet. Animportant to note is that when 4 0, xs will beequal. Furthermore, when

    4 < 0, there will be no

    real roots to the equation under consideration.

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    Application

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    Higher-Degree PolynomialsWe have seen how to deal with polynomials of degree 2

    by making use of the quadratic equation. However,what happens when we encounter higher-degree

    polynomials?

    It turns out that there is a relatively intuitive methodto obtain the roots of higher-degree polynomials.

    If a system reduces to an Nth degree polynomialequation, it will probably have N roots.

    Can be found on Page 38-40 of textbook.

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    Dealing with Higher-degree

    Polynomials Given polynomial equation with integer coefficients:

    + + + + 0

    If there exist rational roots r/s where r, s are integers withno common divisor except unity, then r is a divisor of and s is a divisor of .

    Note there are three theorems listed in the textbook.Theorem 1 is just a subset of theorem 2 whilst theorem 3 issimply a special case scenario.

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    Example Given: +

    10 + 6 0

    First, get rid of the fractions:

    2+ 5 11 20 + 12 0 r={1;-1;2;-2;3;-2;4;-4;6;-6;12;-12} s={1;-1;2;-2}

    r/s={1;-1;1/2;-1/2;2;-2;3;-3;3/2;-3/2;4;-4;6;-6;12;-12}

    Once you have calculated each root, sub each into theoriginal equation and observe which roots satisfy theequation.

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    Final note on polynomials If 2 is a root, then (x-2) is a term. Similarly, if -6 was a root, then

    (x+6) would also be a term. Since the polynomial is of degree 3,we should typically have 3 roots. However, this does not precludethe same roots occurring twice in our final answer.

    Note, in the special case that the coefficients add to zero, 1 is aroot.

    Do the example yourselves and check that you get the followinganswer:

    ( )( 2)( + 2)( + 3) 0

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    Unique Solution? There are a few issues that can confound the

    obtainment of a unique solution.

    We can have inconsistency:

    X+Y=2 and X+Y=3

    Functional dependence:

    X+Y=2 and 2X+2Y=4

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    What Next? So, we have dealt with a system of equations and

    shown how to solve them. This has been relatively easyup to now.

    However, consider a very large system of equations.Writing the whole system out and solving via the

    simultaneous equation method can become verymessy very quickly.

    So, how would we approach this?

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    Chapter 4

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    MatricesA matrix is an array of elements.

    Matrices are a useful tool to summarise a system ofequations when it seems too tedious to write out.

    For example: 2X+5Y=39 & 11X-3Y=1

    It turns out that weare also able to solve

    the system ofequations viamatrices too. We willdiscuss this later.

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    Matrices Contd. Convention dictates that notation be written with the rows

    expressed first followed by the column dimension. Forexample: A (2 x 3) Matrix would have 2 rows and 3

    columns.

    In general, a matrixwith m rows and n columns is an (mx n) matrix.

    Square matrices have the same number of rows as columns.

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    Matrices - General Form

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    Matrices Contd.A matrix containing only one row or one column are

    sometimes referred to as row and column vectorsrespectively.

    General Notation: A=[] Note that the row component always comes first in

    matrices. Therefore, the i subscript relates to the row

    element whereas, j subscript refers to the columnelement.

    A=[]would refer to the element that corresponds tothe second row and and first column.

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    Matrix Operations - Addition

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    Matrix Operations - Subtraction

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    Scalar Matrix Multiplication

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    Matrix Operations - Multiplication

    Much more sophisticated than usual multiplication ofscalars.

    In matrices, compatibility issues arise. One needs tocheck that the dimensions of the matrices allow for the

    matrix multiplication operation to take place.

    Therefore, matrices are order sensitive!

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    Matrix Multiplication Operation

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    NOTE: Always Check Dimensions!

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    A Nifty Trick for Multiplication

    A

    B

    AB

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    Matrix Operations - DivisionWhile matrices are able to perform the normal

    operations that one is typically used to, but it cannotundergo division.

    HOWEVER, there are other operations such asInverses or Reciprocal that takes its place. .

    Aside for later: Transposes are also frequently used.

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    A note on Summation Summation is self-explanatory in the sense that we

    simply the different terms of interest.

    Some people get confused with the shorthandnotation for summation though.

    For example: = This simply indicates that the variable must be

    summed from zero to five. Simply replace i with therange of numbers in the summation operator.

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    Linear Dependence In words: A set of vectors is said to be linearly

    dependentif and only if any one of them can be

    expressed as a linear combination of the remaining

    vectors; otherwise they are linearly independent.

    = 0

    If this equation can be satisfied only when 0forall , then these vectors are said to be linearlyindependent.

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    Identity Matrix Square Matrix with 1s in its principal diagonal and os

    everywhere else.

    Plays a role similar to 1. In other words, any matrixmultiplied by the identity matrix will yield the samematrix given that they are conformable.

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    Null MatrixA matrix all of whose elements are zero.

    Plays the role of the number zero for matrices. Anymatrix multiplied by the null matrix will yield the nullmatrix.

    Note: This matrix does not have to be a square matrix.