__Random variable__ – variable that takes on numerical values determined by the outcome of a random experiment. Usually denoted by X

__Probability distribution__ – expresses probability that a continuous random variable takes on value X=x

- Adding a constant to a variable shifts the distribution
- Multiplying by a constant changes the dispersion of a variable (if <1σ reduces, if >1σ increases)

To compute the probability for a normal random variable, convert X~N(μ, σ^{2}) to Z~N(μ, σ^{2}):

__Point Prediction__ – If data is independent, the best prediction is the mean. If data is not independent (runs test significant at <0.05) then X_{t+1 }= X_{t} + μ_{change} + ε_{t}

**Runs Test **

- Few runs -> cyclical time series
- Many runs -> oscillating time series
- Medium # of runs -> independent time series
- When we predict for independent series (=μ of series) the confidence interval is a multiple of σ
- When we predict for non-independent time series (=μ
_{change}+ ε_{t}) the confidence interval is a multiple of the variance (σ^{2}) {e.g. X_{70}~ N(X_{68}+ 2μ_{change}, 2σ_{change}^{2})}

A **random walk** exists is the changes from period to period are independent and normally distributed

Estimating μ and σ {X representative of μ, s^{2} representative of σ^{2})

- If X ~ N(μ, σ
^{2}) then X ~ N(μ, σ^{2}/n)

Linear Regression (α + ßx) + ε_{t} with ε_{t }iid N(μ, σ_{ε}^{2})

Make distance from each point to the estimated line as small as possible. Minimize Σε_{t}^{2}

4 Basic Assumptions and how to check them:

- linearity – non-linear pattern in x/y plot; u-shaped pattern in the Residuals vs. Fit plot
- constant variance – look for funnel in the Residuals vs. Fit plot
- independence – pattern in time series plot; runs test
- normality – histogram of e’s (residual = e)

Assuming X is independent (data set) implies the e’s are independent. Must check this.

Specification bias – exists when an explanatory variable that should be included in regression is left out. {if a coefficient doesn’t make sense it could be due to specification bias}

Understanding the Regression Output:

In regressions, error terms don’t accumulate (predictions)

Coefficients are random variables because they depend on sample of points

Any time you standardize using an estimate you get a t-distribution not a z-distribution

- Dummy variables are useful when the # of data points isn’t sufficient for multiple single regressions
- Interpret regression model using dummy variables as multiple parallel regression lines where the coefficients are the distance between the lines
- Baseline is the dummy variable that is left out (intentionally) when running the regression
- Best prediction of true regression α + ßx = ε is a + bx (ε=0)

Confidence Intervals

2 components of prediction error:

- (α + ßx) – (a + bx)
- ε
- the ore uncertainty about the 2 sources of error, the more uncertain the prediction is and therefore a wider confidence interval
- Minitab gives a 95% interval

Time Series with Trend

Pattern consists of:

- Trend – model with Time variable
- Seasonal component – model with multiple dummy variables
- Short-term correlation (cyclical) – model w/ Y
_{t-1} - Unpredictable component (ε)

To modify the model to account for non-constant variance, run a regression model for % change in variable (i.e. sales) {this is like a non-independent time-series point prediction}

Leading indicators can improve the model *BUT* are not useful for predicting

It is important to know how close b is to ß:

The sampling distribution for b is b ~ N(ß, σ_{b}^{2})

Hypothesis Testing: (H_{0}: ß=0 H_{A}: ß<>0) – useful for determining which variables have explanatory power

Type I Error – say ß=1 when ß=0 (reject H_{0} when H_{0} is true)

Type II Error – say ß=0 when ß=1 (accept H_{0} when H_{0} is false)

Procedure:

- Collect sample
- Compute b
- Compare to critical t-value
- Reject H
_{0}if (b/s_{b}) > t_{critical}

We’re choosing a cutoff value such that pr(Type I error) = 0.05 (α)

Each coefficient has a t-distribution with (n – total # coefficients) degrees of freedom (t_{n-#}, _{α})

- If one-sided decision (i.e. H
_{A}: ß>0), use α as given percentage - If two-sided decision (i.e. H
_{A}: ß>0), use α as ½ of the given percentage