tailieunhanh - Lecture Introductory econometrics for finance – Chapter 2: Mathematical and statistical foundations
This chapter presents the following content: Straight lines, plot of hours studied against mark obtained, quadratic functions, the roots of quadratic functions, calculating the roots of quadratics, manipulating powers and their indices, logarithms, how do logs work?,. | Chapter 2 Mathematical and Statistical Foundations ’Introductory Econometrics for Finance’ c Chris Brooks 2013 1 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 (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 + 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’ c Chris Brooks 2013 2 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 + • The intercept, a, is 25 and the slope, b, is • 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 , so another 100 hours of study would lead to a 5% increase in the mark ’Introductory Econometrics for Finance’ c Chris Brooks 2013 3 Plot of Hours Studied Against Mark Obtained ’Introductory Econometrics for Finance’ c Chris Brooks 2013 4 Straight Lines • In the graph above, the slope is positive – . 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 – . 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 =
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