Summary
Samenvatting - Quantitative Research Methods III
- Course
- Institution
Complete summary of all the lecture videos and book chapters of QRM3 (Finance & economics) given in year 3 of the Study Economics and Business economics
[Show more]
1 review
By: achbfull • 8 months ago
Seller
Follow
noahebbes
Reviews received
Content preview
Week 1a1 Math/Statistics refresherMath refresher
Functions part 1: linear and quadratic functions, polynomials, working with powers
Linear functions:
A function is a relationship between an input or set of inputs and an output. We write that y = f(x).
If the equation is linear, we have 𝑦 = 𝑏0 + 𝑏1𝑥. where y and x are called variables and b0 and b1
are parameters.
Quadratic functions:
A linear function is often not sufficiently flexible to accurately describe the relationship between two
series. A polynomial adds higher order powers of x into the function
2 𝑛
𝑦 = 𝑏0 + 𝑏1𝑥 + 𝑏2𝑥 + 𝑏33 +. . . + 𝑏𝑛𝑥 . Setting n = 2 is sufficient for many cases
2
𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐
where a, b, c are the parameters that describe the shape of the function. If a is positive, the function
will be ∪-shaped, while if a is negative it will be ∩-shaped.
The Roots of Quadratic functions:
A quadratic equation has two roots. The roots can be obtained either by factoring the equation
(contracting it into parentheses), or by using the
2
−𝑏 ± 𝑏 −4𝑎𝑐
abc-formule: 𝑥 = 2𝑎
2
- 𝑦 = 𝑥 − 4𝑥 > 𝑥=0 ∨𝑥=4
2
- 𝑦 = 𝑥 + 𝑥 − 6 > 𝑥= −3 ∨𝑥=2
2
- 𝑦 = 𝑥 − 3𝑥 + 1 > here factoring doesn’t work, so use the abc-formula
2 2
−(−3) − (−3) −4*1*1 3− 5 −(−3) + (−3) −4*1*1 3+ 5
𝑥 = 2*1
= 2
v𝑥 = 2*1
= 2
Powers of number or of variables:
3
𝑥 = 𝑥 · 𝑥 · 𝑥, here 3 we call the index
Seven frequently used rules:
0
- 𝑥 =1 everything to the power 0 is equal to 1
−2 1
- 𝑥 = 2
𝑥
3 3 3
- (𝑥𝑦) = 𝑥 𝑦
3
2 3 5 𝑥
- 𝑥 ·𝑥 =𝑥 2 =𝑥
𝑥
2 3 6
- (𝑥 ) = 𝑥
𝑛
𝑥 𝑛 𝑥
- (𝑦) = 𝑛
𝑦
1/2 1/𝑛 𝑛
- 𝑥 = 𝑥 𝑥 = 𝑥
Functions part 2: exponential/logarithmic functions
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
𝑥
current value, we would write 𝑦 = 𝑒 with e a simple number: 2.71828
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 Consider the power
3
relationship 2 = 8
Using logarithms, we would write this as 𝑙𝑜𝑔2 8 = 3, or ‘the log to the base 2 of 8 is 3’
𝑏
More generally, if 𝑎 = 𝑐, then we can also write 𝑙𝑜𝑔𝑎 𝑐 = 𝑏
A log to base e is known as a Natural logarithm, denoted interchangeably by ln(y) or log(y). Taking a
natural logarithm is the inverse of taking an exponential, so sometimes the exponential function is
called the antilog.
For variables x and y:
- 𝑙𝑛(𝑥𝑦) = 𝑙𝑛 𝑥 + 𝑙𝑛 𝑦 and 𝑙𝑛(𝑥/𝑦) = 𝑙𝑛 𝑥 − 𝑙𝑛 𝑦
𝑛
- 𝑙𝑛(𝑥 ) = 𝑛 𝑙𝑛 𝑥
- 𝑙𝑛(1) = 0 and 𝑙𝑛(𝑒) = 1
- 𝑙𝑛(1/𝑦) = 𝑙𝑛(1) − 𝑙𝑛(𝑦) = − 𝑙𝑛(𝑦)
𝑥 𝑙𝑛(𝑥)
- 𝑙𝑛(𝑒 ) = 𝑒 =𝑥
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 instance, we might
4
write ∑ 𝑥𝑖 where the i subscript is an index, 1 is the lower limit and 4 is the upper limit of the sum.
𝑖=1
This would mean adding all of the values of x from 𝑥1 𝑡𝑜 𝑥4
Properties of the Sigma operator:
𝑛 𝑛 𝑛
- ∑ 𝑥𝑖 + ∑ 𝑧𝑖 = ∑ (𝑥𝑖 + 𝑧𝑖)
𝑖=1 𝑖=1 𝑖=1
𝑛 𝑛
- ∑ 𝑐𝑥𝑖 = 𝑐 ∑ 𝑥𝑖
𝑖=1 𝑖=1
𝑛 𝑛 𝑛
- ∑ 𝑥𝑖𝑧𝑖 ≠ ∑ 𝑥𝑖 ∑ 𝑧𝑖
𝑖=1 𝑖=1 𝑖=1
𝑛
- ∑ 𝑥 = 𝑥 + 𝑥 +... + 𝑥 = 𝑛𝑥
𝑖=1
𝑛
- ∑ 𝑥 = 𝑥1+ 𝑥2 + ... + 𝑥𝑛 = 𝑛𝑥
𝑖=1
Example:
4
2 2 2 2
∑ (𝑗 + 𝑗 + 3) = (2 + 2 + 3) + (3 + 3 + 3) + (4 + 4 + 3) = 47
𝑗=2
Pi notation:
Similar to the use of sigma to denote sums, the pi operator ( ∏ ) is used to denote repeated
𝑛
multiplications. For example ∏ 𝑥𝑖 = 𝑥1𝑥2. . . 𝑥𝑛
𝑖=1
,means ‘multiply together all of the xi for each value of i between the lower and upper limits.’ It also
follows that
𝑛 𝑛
𝑛
∏ (𝑐𝑥𝑖) = 𝑐 ∏ 𝑥𝑖
𝑖=1 𝑖=1
Example:
6
∏ 𝑖 = 3 * 4 * 5 * 6 = 360
𝑖=3
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 Consider a variable y that is a function f of another variable x, i.e. y = f (x): the
derivative of y with respect to x is written:
𝑑𝑦 𝑑𝑓(𝑥)
𝑑𝑥
= 𝑑𝑥
or 𝑓'(𝑥)
For non-linear functions, the gradient at a certain point is tangent at that point.
Differentiation: the basics:
The derivative of a power function n of x:
𝑛 𝑑𝑦 𝑛−1
if 𝑦 = 𝑐𝑥 then 𝑑𝑥
= 𝑐𝑛𝑥
The derivative of the log of x is given by 1/x:
𝑑(𝑙𝑜𝑔(𝑥)) 1
𝑑𝑥
= 𝑥
𝑥 𝑥
The derivative of 𝑒 is 𝑒
Four rules for derivatives:
- The derivative of a sum is equal to the sum of the derivatives of the individual parts:
𝑑𝑦
𝑦 = 𝑓(𝑥) + 𝑔(𝑥) then 𝑑𝑥
= 𝑓'(𝑥) + 𝑔'(𝑥)
- The derivative of a product of two functions f (x)g(x) is given by
𝑑𝑦
𝑑𝑥
= 𝑓'(𝑥)𝑔(𝑥) + 𝑓(𝑥)𝑔'(𝑥)
𝑓(𝑥)
- The derivative of a quotient of two functions 𝑔(𝑥)
is given by
𝑑𝑦 𝑓'(𝑥)𝑔(𝑥)−𝑔'(𝑥)𝑓(𝑥)
𝑑𝑥
= 2
𝑔(𝑥)
- Suppose we would like to differentiate a function of a function, 𝑦 = 𝑓(𝑔(𝑥)). Then the chain
rule says:
𝑑𝑦 𝑑𝑦 𝑑𝑔
𝑑𝑥
= 𝑑𝑔 𝑑𝑥
Higher order derivatives:
It is possible to differentiate a function more than once to calculate the second order, third order
derivatives. The notation for the second order derivative, which is usually just termed the second
derivative, is
2 𝑑𝑦
𝑑𝑦 𝑑( 𝑑𝑥 )
2 = 𝑓''(𝑥) = 𝑑𝑥
𝑑𝑥
The second order derivative can be interpreted as the gradient of the gradient of a function – i.e., the
rate of change of the gradient. First and second order derivatives are useful when optimizing
functions! When a function reaches a maximum, its second derivative is negative, while it is positive
for a minimum.
Partial differentiation:
In the case where y is a function of more than one variable it may be of interest to determine the effect
that changes in each of the individual x variables would have on y. (Linear regression models!!!)
, 𝑦𝑖 = β0 + β1𝑥1𝑖 + β2𝑥2𝑖 + β3𝑥3𝑖 + ε𝑖
We calculate these partial derivatives one at a time, treating all of the other variables as if they were
constants.
Statistics refresher
Random variables:
A random variable is any variable whose value cannot be predicted exactly. There are discrete and
continuous random variables.
- discrete: specific set of possible values (events); (e.g. throw a dice). It is a variable with a
countable number of distinct values
- continuous: a continuous range of values (e.g. the temperature) the population is the set of all
possible values of the random variable. A numerical variable that can have any value within
an interval is continuous (e.g., 427.21 grams). Sometimes we round a continuous
measurement to an integer (e.g., 427 grams), but that does not make the data discrete.
Probability distributions:
A probability distribution of a discrete random variable lists all events and the probability that each
value will occur. A cumulative distribution function (cdf) is the probability that the random variable
is less than or equal to a particular value. For continuous random variables: probability distribution
becomes a probability density function (pdf) (or density function, or
density
Example: the Normal distribution!
A discrete PDF shows the probability of each X-value, while the CDF
shows the cumulative sum of probabilities, adding from the smallest to
the largest X- value. The figure illustrates a discrete PDF and the
corresponding CDF. Notice that the CDF approaches 1, and the PDF
values of X will sum to 1.
For sketching a normal distribution:
Exercise R.22:
A scalar multiple of a normally-distributed random variable also has a normal distribution. A random
variable X has a normal distribution with mean 5 and variance 10. Sketch the distribution of Z = X / 2.
Answer:
The mean and variance for random variable X are given (5 & 10) and also the relationship between Z
and X (Z = X/2)
𝑋 1 1
So, mean of Z is: 𝐸(𝑍) = µ = 𝐸( 2 ) = 2
𝐸(𝑋) = 2
× 5 = 2. 5
𝑋 1 1
Variance of Z: 𝑉𝑎𝑟(𝑍) = 𝑉𝑎𝑟( ) =
2 4
𝑉𝑎𝑟(𝑋) = 4
× 10 = 2. 5
So, 𝑍 ∼ 𝑁(2. 5, 2. 5) when we sketch this the mean (so the middle) of the distribution is on 2.5.
The variance is equal to 2.5 so the graph starts/ends on 2 * 𝑉𝑎𝑟(𝑋) = 2 * 2. 5 = 5. So the start of
the graph is on 2. 5 − 5 =− 2. 5 and the end of the Z distribution is on 2. 5 + 5 = 7. 5
Expected values and variance:
The expected value, expectation, or mean of a random variable Y, 𝐸(𝑌) is the long-run average
values of the random variable Discrete: Suppose Y takes n possible values.
[ ]
- Discrete: suppose Y takes n possible values 𝑦1, 𝑦2,..., 𝑦𝑛 and 𝑃𝑟 𝑌 = 𝑦𝑖 = 𝑝𝑖 then:
𝑛
𝐸(𝑌) = 𝑦1𝑝1 + 𝑦2𝑝2 +... + 𝑦𝑛𝑝𝑛 = ∑ 𝑦𝑖𝑝𝑖
𝑖=1
This is called the population mean
- continuous: probability-weighted average of the possible outcomes of the random variable
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller noahebbes. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $9.96. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
52510 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now