Urgent Econometrics test
wc15920665166
Lecture01.pptxAn Introduction to Econometrics
Econometrics is about:
Studying the relationship between economic variables
Using data to test economic theories. For example, using the appropriate data one can test whether there is positive relationship between household per capita food expenditure and income.
It can be used to quantify the impact of an economic variable on another economic variable. For example, using data one can estimate by how many percent demand for oil drops when price increases by 1%.
What is econometrics about?
The relationships between economic variables can be described by a mathematical function
What is econometrics about?
An econometric model can be written as a combination of deterministic component and a random component (e is also know as random error)
Econometric Model
The deterministic part is derived from economic theory by assuming an appropriate functional form
The random part represents uncertainty. It includes all the unobserved variables.
The econometric model is used for making statistical inference
Econometric Model
Experimental data
Non-experimental data. Example: Survey data
Types of data:
Time series data. Example: GDP and inflation data for a country over time.
Cross-section data. Example: Wage and socio-economic data for individuals (sample broadly representing workers in Canberra) in Canberra in 2015.
Panel data: Example: Wage and socio-economic data for individuals in Canberra during the period: 2000-2015.
Data
Micro or Macro
Flow or a Stock:
Flow: measured over a period of time
Stock: measured at a particular point in time
Quantitative or Qualitative:
Quantitative: expressed as continuous or discrete numbers
Qualitative: binary, ranked or unranked data
Data
Probability Primer
A random variable is a variable which takes values from some distribution. However, the value it takes at particular instance is unknown
Discrete random variable
Continuous random variable
The probability density function (pdf) for a discrete random variable indicates the probability of each possible value occurring
For a discrete random variable X the value of the probability density function f(x) is the probability that the random variable X takes the value x, f(x) = P(X = x)
The following must be true: 0 ≤ f(x) ≤ 1
f(x1) + f(x2) + … + f(xn) = 1
The cumulative density function (cdf ) of the random variable X, denoted F(x), gives the probability that X is less than or equal to a specific value x:
Random Variables & Probability Distributions
Source:Tables P.1 & P.2
A joint probability is about the probability of two events occurring together
Example: In Table P.3
Source: Table P.3 Joint Probability Density Function for X and Y
Joint, Marginal and Conditional Probabilities
If joint probability density function is known then one can derive the probability distributions (also known as marginal distributions) of individual random variables
The probability that Y = 1 is:
Source: Table P.4 Joint and Marginal Probabilities
Marginal Probabilities
The conditional probability is given by the conditional pdf f(x|y):
Example:
Conditional Probabilities
If X and Y are independent random variables, then:
or
X and Y are statistically independent if their joint pdf is the product of their marginal pdf ’s:
Statistical independence
The expected value of random variable X can be written as:
Example:
Expected value of a random variable
For a discrete random variable X the conditional expectation, given Y=y is written as
Conditional Expectations
If g(X) is a function of the random variable X, then g(X) is also random
If X is a discrete random variable, then the expected value of g(X) can be derived using:
If a is a constant, then g(X) = aX is a function of X, and:
If a and b are constants, then:
If g1(X) and g2(X) are functions of X, then:
Variance
Example:
Variance
Source: FIGURE P. 3 Distributions with different variances
If X and Y are two random variables. It then follows:
and
E(XY) = E(X)E(Y) if X and Y are independent
Covariance between two random variable X and Y respectively is defined as
Source: FIGURE P.4 Correlated data
Covariance
If X and Y are measured in different units then interpreting covariance between them is not straightforward
Scaling the covariance by the standard deviations of the variables gets rid of the units of measurement
The correlation always lies between –1 and +1
Correlation
If a and b are constants, then:
If a=1 and b=1 then:
If a=1 and b=-1 then:
If X is a normally distributed random variable with mean μ and variance σ2 , the pdf of X is given by :
A standard normal random variable is one that has a normal probability density function with mean 0 and variance 1
If X ~ N(μ, σ2), then:
Normal distribution
If X ~ N(μ, σ2) and a and b are constants, then:
Normal distribution
Example: If X ~ N(3, 9),then
A weighted sum of normal random variables has a normal distribution
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