Some estimates.

Question. What’s an example? A sporting event?

estimate of the mean level

We will see how to estimate in Chapter 14

3.3 Regression methods.

Unbiased – proof by substitution

= 0

Analysis of varaiance rwalk + drift

std.errors assumed iid Gaussian here

betas are constants

Reminder. The beta’s are fixed (and that the error is

stationary).

Cosine trend.

NB. PHI is assumed constant

NB. –π < atan2(y,x) <= π or in [0,2π)

Yt = β0 + β1 X1t + β2 X2t + Zt

More later

p-value: Prob get a result as or more extreme as observed test statistic

(|t-value| here) under the null model

Fitted curve

3.5 Interpreting output

3.6 RESIDUAL ANALYSIS

Yt = μt + Xt

X̂ = Yt - μ̂t

μt (θ) =̂ μ̂t = μt ( 𝜃)

“most statistics software will produce standardized residuals using a

more complicated standard error in the denominator that takes into

account the specific regression model being fit”

EDA tools.

There are a broad variety of very important residual plots including:

residuals versus fitted values mu-hat transform response?

residuals versus omitted explanatories other X’s include them?

residual versus index i add to model?

residual versus preceding value X[t-1] autocorellation?

qqnorm infer normality?

qqnorm(): displays the quantiles of the data versus the theoretical

quantiles of a normal distribution

Next. transform response?

winter spring/fall summer

rstudent(): standardized residuals .

Could add a line

Sample autocorrelation function.

larain correlogram example

skewed larain stleaf()

the decimal point is at the |

4 | 1168958 6 | 235544566 8 | 01357789991368 10 | 5677790236899 12 | 01367889012677 14 | 133457801334 16 | 02577702455559 18 | 00266778929 20 | 3895 22 | 0672379 24 | 0 26 | 3382 28 | 30 | 63 32 | 3 34 | 0 36 | 38 | 40 | 3

A stochastic (regression) model

Yt = β0 + β1 X1t + β2 X2t + Zt

Zt independent zero mean, white noise, or ar or ma or arma(p,q)

harmonic() creates an [1 X1t X2t ] matrix

classical approach to regression / ordinary least squares

Interpreting regression coefficients.

E{Yt | X1t , X2t} = β0 + β1 X1t + β2 X2t

If increase X1t by 1 keeping X2t fixed, expected value increases by β1

Compare partial derivative ∂∙/∂x1