PowerPoint Presentation

Chapter 1 Introduction and Mathematical Foundations ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 1.

Introduction: The Nature and Purpose of Econometrics What is Econometrics? Literal meaning is “measurement in economics” Definition of financial econometrics: The application of statistical and mathematical techniques to problems in finance. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 2.

[Audio] Examples of the kind of problems that may be solved by an Econometrician 1. Testing whether financial markets are weak form informationally efficient. 2. Testing whether the C-A-P-M or A-P-T represent superior models for the determination of returns on risky assets. 3. Measuring and forecasting the volatility of bond returns. 4. Explaining the determinants of bond credit ratings used by the ratings agencies. 5. Modelling long term relationships between prices and exchange rates. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 3.

[Audio] Examples of the kind of problems that may be solved by an Econometrician (cont’d) 6. Determining the optimal hedge ratio for a spot position in oil. 7. Testing technical trading rules to determine which makes the most money. 8. Testing the hypothesis that earnings or dividend announcements have no effect on stock prices. 9. Testing whether spot or futures markets react more rapidly to news. 10.Forecasting the correlation between the returns to the stock indices of two countries. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 4.

[Audio] What are the Special Characteristics of Financial Data? Frequency & quantity of data Stock market prices are measured every time there is a trade or somebody posts a new quote. Quality Recorded asset prices are usually those at which the transaction took place. No possibility for measurement error but financial data are “noisy”. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 5.

[Audio] Steps involved in the formulation of econometric models Economic or Financial Theory (Previous Studies) Formulation of an Estimable Theoretical Model Collection of Data Model Estimation Is the Model Statistically Adequate? No Yes Reformulate Model Interpret Model Use for Analysis ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 6.

[Audio] Some Points to Consider when reading papers in the academic finance literature 1. Does the paper involve the development of a theoretical model or is it merely a technique looking for an application, or an exercise in data mining? 2. Is the data of “good quality”? Is it from a reliable source? Is the size of the sample sufficiently large for asymptotic theory to be invoked? 3. Have the techniques been validly applied? Have diagnostic tests been conducted for violations of any assumptions made in the estimation of the model? ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 7.

[Audio] Some Points to Consider when reading papers in the academic finance literature (cont’d) 4. Have the results been interpreted sensibly? Is the strength of the results exaggerated? Do the results actually address the questions posed by the authors? 5. Are the conclusions drawn appropriate given the results, or has the importance of the results of the paper been overstated? ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 8.

[Audio] Functions A function is a mapping or relationship between an input or set of inputs and an output We write that y, the output, is a function f of x, the input, or y = f(x) y could be a linear function of x where the relationship can be expressed on a straight line Or it could be non linear where it would be expressed graphically as a curve If the equation is linear, we would write the relationship as y = a plus bx where y and x are called variables and a and b are parameters a is the intercept and b is the slope or gradient ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 9.

[Audio] Straight Lines The intercept is the point at which the line crosses the y axis Example: suppose that we were modelling the relationship between a student’s average mark, y (in percent), and the number of hours studied per year, x Suppose that the relationship can be written as a linear function y = 25 plus 0.05x The intercept, a, is 25 and the slope, b, is 0.05 This means that with no study (x=0), the student could expect to earn a mark of 25% For every hour of study, the grade would on average improve by 0.05%, so another 100 hours of study would lead to a 5% increase in the mark ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 10.

[Audio] Plot of Hours Studied Against Mark Obtained ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 11.

[Audio] Straight Lines In the graph above, the slope is positive – in other words the line slopes upwards from left to right But in other examples the gradient could be zero or negative For a straight line the slope is constant – in other words the same along the whole line In general, we can calculate the slope of a straight line by taking any two points on the line and dividing the change in y by the change in x  (Delta) denotes the change in a variable For example, take two points x=100, y=30 and x=1000, y=75 We can write these using coordinate notation (x,y) as (100,30) and (1000,75) We would calculate the slope as ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 12.

[Audio] Roots The point at which a line crosses the x axis is known as the root A straight line will have one root (except for a horizontal line such as y=4 which has no roots) To find the root of an equation set y to zero and rearrange 0 = 25 plus 0.05x So the root is x = −500 In this case it does not have a sensible interpretation: the number of hours of study required to obtain a mark of zero! ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 13.

[Audio] Quadratic Functions A linear function is often not sufficiently flexible to accurately describe the relationship between two series We could use a quadratic function instead. We would write it as y = a plus bx plus cx2 where a, b, c are the parameters that describe the shape of the function Quadratics have an additional parameter compared with linear functions The linear function is a special case of a quadratic where c=0 a still represents where the function crosses the y axis As x becomes very large, the x2 term will come to dominate Thus if c is positive, the function will be -shaped, while if c is negative it will be -shaped. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 14.

[Audio] The Roots of Quadratic Functions A quadratic equation has two roots The roots may be distinct (in other words, different from one another), or they may be the same (repeated roots); they may be real numbers (for example, 1.7, -2.357, 4, et cetera) or what are known as complex numbers The roots can be obtained either by factorising the equation (contracting it into parentheses), by ‘completing the square’, or by using the formula: ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 15.

[Audio] The Roots of Quadratic Functions (Cont’d) If b2 > 4ac, the function will have two unique roots and it will cross the xaxis in two separate places If b2 = 4ac, the function will have two equal roots and it will only cross the x axis in one place If b2 < 4ac, the function will have no real roots (only complex roots), it will not cross the x axis at all and thus the function will always be above the x axis. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 16.

[Audio] Calculating the Roots of Quadratics Examples Determine the roots of the following quadratic equations: 1. y = x2 plus x − 6 2. y = 9x2 plus 6x plus 1 3. y = x2 − 3x plus 1 4. y = x2 − 4x ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 17.

[Audio] Calculating the Roots of Quadratics Solutions We solve these equations by setting them in turn to zero We could use the quadratic formula in each case, although it is usually quicker to determine first whether they factorise 1. x2 plus x − 6 = 0 factorises to (x − 2)(x plus 3) = 0 and thus the roots are 2 and −3, which are the values of x that set the function to zero. In other words, the function will cross the x axis at x = 2 and x = −3 2. 9x2 plus 6x plus 1 = 0 factorises to (3x plus 1)(3x plus 1) = 0 and thus the roots are −1/3 and −1/3. This is known as repeated roots – since this is a quadratic equation there will always be two roots but in this case they are both the same. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 18.

[Audio] Calculating the Roots of Quadratics – Solutions Cont’d 3. x2 − 3x plus 1 = 0 does not factorise and so the formula must be used with a = 1, b = −3, c = 1 and the roots are 0.38 and 2.62 to two decimal places 4. x2 − 4x = 0 factorises to x(x − 4) = 0 and so the roots are 0 and 4. All of these equations have two real roots But if we had an equation such as y = 3x2 − 2x plus 4, this would not factorise and would have complex roots since b2 − 4ac < 0 in the quadratic formula. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 19.

[Audio] Powers of Number or of Variables A number or variable raised to a power is simply a way of writing repeated multiplication So for example, raising x to the power 2 means squaring it (in other words, x2 = x × x). Raising it to the power 3 means cubing it (x3 = x × x × x), and so on The number that we are raising the number or variable to is called the index, so for x3, the index would be 3 ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 20.

[Audio] Manipulating Powers and their Indices Any number or variable raised to the power one is simply that number or variable, for example, 31 = 3, x1 = x, and so on Any number or variable raised to the power zero is one, for example, 50 = 1, x0 = 1, et cetera, except that 00 is not defined (in other words, it does not exist) If the index is a negative number, this means that we divide one by that number – for example, x−3 = 1/(x3) = 1/(x×x×x ) If we want to multiply together a given number raised to more than one power, we would add the corresponding indices together – for example, x2 × x3 = x2x3 = x2 plus 3 = x5 If we want to calculate the power of a variable raised to a power (in other words, the power of a power), we would multiply the indices together – for example, (x2)3 = x2×3 = x6 ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 21.

[Audio] Manipulating Powers and their Indices (Cont’d) If we want to divide a variable raised to a power by the same variable raised to another power, we subtract the second index from the first – for example, x3 / x2 = x3−2 = x If we want to divide a variable raised to a power by a different variable raised to the same power, the following result applies: (x / y)n = xn / yn The power of a product is equal to each component raised to that power – for example, (x × y)3 = x3 × y3 The indices for powers do not have to be integers, so x1/2 is the notation we would use for taking the square root of x, sometimes written √x Other, non integer powers are also possible, but are harder to calculate by hand (for example x0:76, x−0:27, et cetera) In general, x1/n = n√x ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 22.

[Audio] The Exponential Function, e It is sometimes the case that the relationship between two variables is best described by an exponential function For example, when a variable grows (or reduces) at a rate in proportion to its current value, we would write y = ex e is a simply number: 2.71828. . . It is also useful for capturing the increase in value of an amount of money that is subject to compound interest The exponential function can never be negative, so when x is negative, y is close to zero but positive It crosses the y axis at one and the slope increases at an increasing rate from left to right. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 23.

[Audio] A Plot of the Exponential Function ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 24.

[Audio] Logarithms Logarithms were invented to simplify cumbersome calculations, since exponents can then be added or subtracted, which is easier than multiplying or dividing the original numbers There are at least three reasons why log transforms may be useful. 1. Taking a logarithm can often help to rescale the data so that their variance is more constant, which overcomes a common statistical problem known as heteroscedasticity. 2. Logarithmic transforms can help to make a positively skewed distribution closer to a normal distribution. 3. Taking logarithms can also be a way to make a non linear, multiplicative relationship between variables into a linear, additive one. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 25.

[Audio] How do Logs Work? Consider the power relationship 23 = 8 Using logarithms, we would write this as log28 = 3, or ‘the log to the base 2 of 8 is 3’ Hence we could say that a logarithm is defined as the power to which the base must be raised to obtain the given number More generally, if ab = c, then we can also write logac = b If we plot a log function, y = log(x), it would cross the x axis at one – see the following slide It can be seen that as x increases, y increases at a slower rate, which is the opposite to an exponential function where y increases at a faster rate as x increases. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 26.

[Audio] A Graph of a Log Function ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 27.

[Audio] How do Logs Work? Natural logarithms, also known as logs to base e, are more commonly used and more useful mathematically than logs to any other base A log to base e is known as a natural or Naperian logarithm, denoted interchangeably by ln(y) or log(y) Taking a natural logarithm is the inverse of a taking an exponential, so sometimes the exponential function is called the antilog The log of a number less than one will be negative, for example ln(0.5) ≈ −0.69 We cannot take the log of a negative number – So ln(−0.6), for example, does not exist. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 28.

[Audio] The Laws of Logs For variables x and y: ln (x y) = ln (x) plus ln (y) ln (x/y) = ln (x) − ln (y) ln (yc) = c ln (y) ln (1) = 0 ln (1/y) = ln (1) − ln (y) = −ln (y) ln(ex) = eln(x) = x ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 29.

[Audio] Sigma Notation If we wish to add together several numbers (or observations from variables), the sigma or summation operator can be very useful Σ means ‘add up all of the following elements.’ For example, Σ(1 plus 2 plus 3) = 6 In the context of adding the observations on a variable, it is helpful to add ‘limits’ to the summation For instance, we might write where the i subscript is an index, 1 is the lower limit and 4 is the upper limit of the sum This would mean adding all of the values of x from x1 to x4. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 30.

[Audio] Properties of the Sigma Operator ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 31.

[Audio] Pi Notation Similar to the use of sigma to denote sums, the pi operator (Π) is used to denote repeated multiplications. For example means ‘multiply together all of the xi for each value of i between the lower and upper limits’. It also follows that ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 32.

[Audio] Differential Calculus The effect of the rate of change of one variable on the rate of change of another is measured by a mathematical derivative If the relationship between the two variables can be represented by a curve, the gradient of the curve will be this rate of change Consider a variable y that is a function f of another variable x, in other words y = f (x): the derivative of y with respect to x is written or sometimes f ′(x). This term measures the instantaneous rate of change of y with respect to x, or in other words, the impact of an infinitesimally small change in x Notice the difference between the notations Δy and dy ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 33.

[Audio] Differentiation: The Basics 1. The derivative of a constant is zero – for example if y = 10, dy/dx = 0 This is because y = 10 would be a horizontal straight line on a graph of y against x, and therefore the gradient of this function is zero 2. The derivative of a linear function is simply its slope for example if y = 3x plus 2, dy/dx = 3 But non linear functions will have different gradients at each point along the curve In effect, the gradient at each point is equal to the gradient of the tangent at that point The gradient will be zero at the point where the curve changes direction from positive to negative or from negative to positive – this is known as a turning point. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 34.

[Audio] The Tangent to a Curve ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 35.

[Audio] The Derivative of a Power Function or of a Sum The derivative of a power function n of x, in other words y = cxn is given by dy/dx = cnxn−1 For example: – If y = 4x3, dy/dx = (4 × 3)x2 = 12x2 – If y = 3/x = 3x−1, dy/dx= (3 × −1)x−2 = −3x−2 = −3/x2 The derivative of a sum is equal to the sum of the derivatives of the individual parts: for example, if y = f (x) plus g (x), dy/dx = f ′(x) plus g′(x) The derivative of a difference is equal to the difference of the derivatives of the individual parts: for example, if y = f (x) − g (x), dy/dx = f ′(x) − g′(x). ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 36.

[Audio] The Derivatives of Logs and Exponentials The derivative of the log of x is given by 1/x, in other words d(log(x))/dx = 1/x The derivative of the log of a function of x is the derivative of the function divided by the function, in other words d(log(f (x)))/dx = f ′(x)/f (x) E.g., the derivative of log(x3 plus 2x − 1) is (3x2 plus 2)/(x3 plus 2x − 1) The derivative of ex is ex. The derivative of e f (x) is given by f ′(x)e f (x) E.g., if y = e3x2, dy/dx = 6xe3x2 ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 37.

[Audio] Higher Order Derivatives It is possible to differentiate a function more than once to calculate the second order, third order, . . ., nth order derivatives The notation for the second order derivative, which is usually just termed the second derivative, is To calculate second order derivatives, differentiate the function with respect to x and then differentiate it again For example, suppose that we have the function y = 4x5 plus 3x3 plus 2x plus 6, the first order derivative is ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 38.

[Audio] Higher Order Derivatives (Cont’d) The second order derivative is The second order derivative can be interpreted as the gradient of the gradient of a function – in other words, the rate of change of the gradient How can we tell whether a particular turning point is a maximum or a minimum? The answer is that we would look at the second derivative When a function reaches a maximum, its second derivative is negative, while it is positive for a minimum. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 39.

[Audio] Maxima and Minima of Functions Consider the quadratic function y = 5x2 plus 3x − 6 Since the squared term in the equation has a positive sign (in other words, it is 5 rather than, say, −5), the function will have a ∪-shape rather than an ∩shape, and thus it will have a minimum rather than a maximum: dy/dx = 10x plus 3, d2y/dx2 = 10 Since the second derivative is positive, the function indeed has a minimum To find where this minimum is located, take the first derivative, set it to zero and solve it for x So we have 10x plus 3 = 0, and x = −3/10 = −0.3. If x = −0.3, y is found by substituting −0.3 into y = 5x2 plus 3x − 6 = 5 × (−0.3)2 plus (3 × −0.3) − 6 = −6.45. Therefore, the minimum of this function is found at (−0.3,−6.45). ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 40.

[Audio] Partial Differentiation In the case where y is a function of more than one variable (for example y = f (x1, x2, . . . , xn)), it may be of interest to determine the effect that changes in each of the individual x variables would have on y Differentiation of y with respect to only one of the variables, holding the others constant, is partial differentiation The partial derivative of y with respect to a variable x1 is usually denoted ∂y/∂x1 All of the rules for differentiation explained above still apply and there will be one (first order) partial derivative for each variable on the right hand side of the equation. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 41.

[Audio] How to do Partial Differentiation We calculate these partial derivatives one at a time, treating all of the other variables as if they were constants. 2, the partial To give an illustration, suppose y = 3x1 3 plus 4x1 − 2x2 4 plus 2x2 2 plus 4, while the derivative of y with respect to x1 would be ∂y/∂x1 = 9x1 partial derivative of y with respect to x2 would be ∂y/∂x2 = −8x2 3 plus 4x2 The ordinary least squares (O-L-S--) estimator gives formulae for the values of the parameters that minimise the residual sum of squares, denoted by L The minimum of L is found by partially differentiating this function and setting the partial derivatives to zero Therefore, partial differentiation has a key role in deriving the main approach to parameter estimation that we use in econometrics. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 42.

[Audio] Integration Integration is the opposite of differentiation If we integrate a function and then differentiate the result, we get back the original function Integration is used to calculate the area under a curve (between two specific points) Further details on the rules for integration are not given since the mathematical technique is not needed for any of the approaches used here. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 43.

[Audio] Matrices Background Some useful terminology: – A scalar is simply a single number (although it need not be a whole number – for example, 3, −5, 0.5 are all scalars) – A vector is a one dimensional array of numbers (see below for examples) – A matrix is a two dimensional collection or array of numbers. The size of a matrix is given by its numbers of rows and columns Matrices are very useful and important ways for organising sets of data together, which make manipulating and transforming them easy Matrices are widely used in econometrics and finance for solving systems of linear equations, for deriving key results, and for expressing formulae. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 44.

[Audio] Working with Matrices The dimensions of a matrix are quoted as R × C, which is the number of rows by the number of columns Each element in a matrix is referred to using subscripts. For example, suppose a matrix M has two rows and four columns. The element in the second row and the third column of this matrix would be denoted m23. More generally mij refers to the element in the ith row and the jth column. Thus a 2 × 4 matrix would have elements If a matrix has only one row, it is a row vector, which will be of dimension 1 × C, where C is the number of columns, for example (2.7 3.0 −1.5 0.3) ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 45.

[Audio] Working with Matrices A matrix having only one column is a column vector, which will be of dimension R× 1, where R is the number of rows, for example When the number of rows and columns is equal (in other words R = C), it would be said that the matrix is square, for example the 2 × 2 matrix: A matrix in which all the elements are zero is a zero matrix. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 46.

[Audio] Working with Matrices 2 A symmetric matrix is a special square matrix that is symmetric about the leading diagonal so that mij = mji ∀ i, j, for example A diagonal matrix is a square matrix which has non zero terms on the leading diagonal and zeros everywhere else, for example ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 47.

[Audio] Working with Matrices 3 A diagonal matrix with 1 in all places on the leading diagonal and zero everywhere else is known as the identity matrix, denoted by I, for example The identity matrix is essentially the matrix equivalent of the number one Multiplying any matrix by the identity matrix of the appropriate size results in the original matrix being left unchanged So for any matrix M, 1001 = IM = M In order to perform operations with matrices , they must be conformable The dimensions of matrices required for them to be conformable depend on the operation. ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 48.

[Audio] Matrix Addition or Subtraction Addition and subtraction of matrices requires the matrices concerned to be of the same order (in other words to have the same number of rows and the same number of columns as one another) The operations are then performed element by element ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 49.

[Audio] Matrix Multiplication Multiplying or dividing a matrix by a scalar (that is, a single number), implies that every element of the matrix is multiplied by that number More generally, for two matrices A and B of the same order and for c a scalar, the following results hold – A plus B = B plus A – A plus 0 = 0 plus A = A – cA = Ac – c(A plus B) = cA plus cB – A0 = 0 ampere = 0 ‘Introductory Econometrics for Finance’ © Chris Brooks 2019 50.