Transcript Time Series, Cross Sectional & Pooled Data

Time Series, Cross Sectional & Pooled Data
• 1. Time Series Data
-One location’s data across time
-Yearly, monthly, quarterly (every three
months), weekly, daily, etc.
-ie: Canadian GDP, Enron stock value,
your height, U of A tuition, world pop.
Time Series, Cross Sectional & Pooled Data
• 2. Cross-Sectional Data
-Multiple Locations at one time
-Taken at same time (September report,
January report, etc.)
-ie: stock portfolio, player stats,
provincial GDP comparison, grade
report
Time Series, Cross Sectional & Pooled Data
• 3. Pooled Data
-Combination of Time Series and Crosssectional Data
-More difficult to use
-Often required due to data restrictions
General Equations
Nominal Value =
(Price Index/100) X Real value
Or
Real value =
Nominal value / (Price Index/100)
1.2.2 Laspeyres Price Index
-uses base year quantities as weights
-still = 100 in base year
Lt = ∑ pricest X quantitiesbase year
---------------------------------∑ pricesbase year X quantitiesbase year
-tracks cost of buying a fixed (base year)
basket of goods (ie: CPI)
1.2.2 Paasche Price Index
-uses current year quantities as weights
-still = 100 in base year
Pt = ∑ pricest X quantitiest
---------------------------------∑ pricesbase year X quantitiest
-compares cost of current basket now to
cost of current basket in base year
1.2.4 Nominal, Relative, and Real
Price Indexes
Nominal Price Index –
-price index for a good or service
-describes movement of prices over
time
ie: education, gas, coffee
Note: CPI (consumer price index) for all
goods is used to measure inflation
1.2.4 Nominal, Relative, and Real
Price Indexes
Relative Price Index –
-price index for a good or service
relative to another
-describes movement of prices over
time compared to another good or
service
Relative Price Index = Price Index A
----------------------------------
Price Index B
1.2.4 Nominal, Relative, and Real
Price Indexes
Real Price Index –
-price index for a good or service
relative to all others
-describes movement of prices over
time compared to all other goods
Real Price Index = Price Index A
----------------------------------
CPI (all goods)
1.4.2.1 Easy Interest Formula
rreal = (1+rnom-1-inf)
---------------(1+inf)
rreal+ rreal*inf = rnom-inf
(rreal*inf is small)
rreal = rnom – inf
Last example: rreal = 2%-3%=-1%
1.4.3.2 More Frequent Compounding
If interest is compounded m times a year, 1/m
of the interest is paid each time
Modified Formula:
S = P (1+[i/m])mt
S = value after t years P = principle amount
i = interest rate
t = years
m = times compounded (monthly = 12, etc)
Infinite Compounding:
S = Peit
1.4.3.3 Effective Rate of Interest
Which is the better investment: 25% compounded
annually or 24% compounded monthly?
iE = effective rate of interest if
compounded annually
P (1+iE)t = P (1+[i/m])mt
Solving for iE, we get:
iE = (1+[i/m])m-1
1.4.3.4 Present Value
How much do I have to invest now to have a
given sum of money in the future?
PV = S/[(1+i)t]
PV = present value (money invested now)
S = sum needed in future
i = real, compound interest rate
t = years
1.4.3.4 Continued Deposits
How does this change if it’s more than a onetime investment/payment?
(ie: $100 per year for 5 years, 7% interest)
PV= 100+100/1.07 + 100/1.072 + 100/1.073
+ 100/1.074
= 100 + 93.5 + 87.3 + 81.6 + 76.3 = $438.7
Or
PV = a[1-(1/{1+i})t] / [1- (1/{1+i})]
PV = a[1-xt] / [1-x]
x=1/{1+i}
PV = 100[1-(1/1.07)5]/[1-1/1.07] = $438.72
1.5.1 -Stocks and Flows Summary
Type of
Stock
Flow
Variable
Major
Measured at a point in Measured over a
Characteristic time
period (between
points in time)
Examples
Debts, wealth,
housing, stocks,
capital, tuition
Aggregation
Method
Average or
Use values from the
same time each year
Deficits, income,
building starts,
investment,
payments
Sum
(Average if
annualized)
1.5.2 – The User Cost of Capital
User cost of capital = implicit rental rate
=
Pkt ( d + r - [Pkt+1 – Pkt]/Pkt )
d = depreciation – increases rental cost
(more willing to rent a costly item)
r = return on alternate investments
(more willing to rent given high returns)
[Pkt+1 – Pkt]/Pkt = capital gains/losses
(less willing to rent a constant value item)
2.1.4 – Growth Models
The most common formulas to measure
growth are:
1) [{Xt-Xt-1}/Xt-1] X 100
2) [ln(Xt)-ln(Xt-1)] X 100
3) [{dX/dt}/X] X 100
4) [dln(X)/dt] X 100
-1 and 2 work well with data
-3 and 4 require calculus
-any can be used with formulas
2.A.1.2 Rules of Derivatives
-although first principles always work, the
following rules are more economical:
1)Constant Rule
If f(x)=k (k is a constant),
f ‘(x) = 0
2) General Rule
If f(x) = ax+b (a and b are constants)
f ‘ (x) = a
2.A.1.2 Rules of Derivatives
3) Power Rule
If f(x) = kxn,
f ‘(x) = nkxn-1
4) Addition Rule
If f(x) = g(x) + h(x),
f ‘(x) = g’(x) + h’(x)
2.A.1.2 Rules of Derivatives
5) Product Rule
If f(x) =g(x)h(x),
f ‘(x) = g’(x)h(x) + h’(x)g(x)
-order doesn’t matter
6) Quotient Rule
If f(x) =g(x)/h(x),
f ‘(x) = {g’(x)h(x)-h’(x)g(x)}/{h(x)2}
-order matters
-derived from product rule
2.A.1.2 Rules of Derivatives
5) Product Rule
If f(x) =(12x+6)x3
f ‘(x) = 12x3 + (12x+6)3x2
= 48x3 + 18x2
6) Quotient Rule
If f(x) =(12x+1)/x2
f ‘(x) = {12x2 – (12x+1)2x}/x4
= [-12x2-2x]/x4
= [-12x-2]/x3
2.A.1.2 Rules of Derivatives
7) Power Function Rule
If f(x) = [g(x)]n,
f ‘(x) = n[g(x)]n-1g’(x)
-special case of the chain rule
8) Chain Rule
If f(x) = f(g(x)), let y=f(u) and u=g(x), then
dy/dx = dy/du X du/dx
2.A – More Derivatives
1) Natural Logs
If y=ln(x),
y’ = 1/x
-chain rule may apply
If y=ln(x2)
y’ = (1/x2)2x = 2/x
2.A – More Derivatives
2) Trig. Functions
If y = sin (x),
y’ = cos(x)
If y = cos(x)
y’ = -sin(x)
-Use graphs as reminders
2.2 Mathematical Models of Economic
Relationships
Consumption Function – slope = mpc
Consumption = 100+0.5income
Mpc = dc/di = 0.5
Consumption = 100+0.5income-0.02income2
Mpc = dc/di = 0.5-0.04income
Are any other functional forms viable for
consumption?
2.A Second Derivatives
x=15-10t+t*t
10
5
0
x
In this case,
we see the slope
increasing as t
increases,
transitioning
from a negative
slope to a positive
slope.
-5
1
2
3
4
5
6
7
8
-10
-15
t
A second derivative would discover this fact, aid in
the sketching of the graph, and confirm a minimum
point on the graph. Here x’’ is positive.
x
2.A – Implicit Differentiation Rules
1) Take the derivative of EACH term on both
sides.
2) Differentiate y as you would x, except that
every time you differentiate y, multiply that
term by dy/dx (or y’)
Ie: 14=7x+9x2-y
d(14)/dx=d(7x)/dx+d(9x2)/dx-dy/dx
0 = 7 + 18x – y’
y’=7+18x
2.A.1.3 Derivative Applications - Graphs
Graphing Steps:
i) Evaluate f(x) at x=0, ∞, - ∞, or a variety of
values
ii) Determine where f(x)=0
iii) Calculate slope - f ’(x) - and determine
where it is positive and negative
iv) Identify possible maximum and minimum
co-ordinates where f ‘(x)=0. (Don’t just find
the x values)
2.A.1.3 Derivative Applications - Graphs
Graphing Steps:
v) Calculate the second derivative – f ‘’(x) and
use it to determine max/min in iv
vi) Using the second derivative, determine the
curvature (concave or convex) at other
points
vii) Check for inflection points where f ‘’(x)=0
2.2.2 Elasticities
-to avoid this problem, economists often utilize
ELASTICITIES
-elasticities deal with PERCENTAGES and are
therefore more useful across a variety of
ranges
ELASTICITY = a PROPORTIONAL change in y
from a PROPORTIONAL change in x
Example: elasticity of demand:
η = Δy/y / Δx/x
= (Δy/Δx) (x/y)
= (dy/dx) (x/y)
2.2.2 Elastic Logs
The MAIN reason to use logs in economic
formulae is to more easily calculate
elasticities:
E = dy/dx * y/x
= (1/y) dy/dx (x)
= (dlny/dx) dy/dx (dx/dlnx)
= (dlny/dlnx) dx/dx (dy/dy)
= (dlny/dlnx)
2.3 Interpreting Parameters
Unfortunately for economists, our
employers are not awed and amazed
by elegant equations – they want to
know what the elegant equations mean.
Intercepts, slopes, curvature and elasticity
are thus far tools to explain models.
Parameter explanation is what employers
want.
2.3.1 Simple Example
Let mark = 60 + 4 study
Mark = percentage mark on midterm
Study = hours of study (up to 10 – it’s the
night before)
Parameter Explanation:
60 = intercept – without studying, you’d get
a 60% on the exam, you genius you
4 = coefficient of study – every extra hour
spent studying increases your mark by
4%
2.A The Partial Derivative
It is often impossible to analyze all various
movements of explanatory variables and
their impact on the dependent variable.
Instead, we analyze one variable’s impact,
assuming ALL OTHER VARIABLES REMAIN
CONSTANT
We do this through the partial derivative.
2.4 The error term
Although economists try to model real behavior,
their attempts are not always 100% accurate,
for a variety of reasons:
1)
2)
3)
4)
Excluded variables
Random events (shocks)
Error in data collection
Economist stayed up late watching the
hockey game (which could cause the
above)
2.A Optimizing
There are three steps for optimization:
1) Find where f’(x)=0. This is your FIRST
ORDER CONDITION (FOC) and gives
potential maxima/minima.
2) Evaluate f’’(x) at your potential
maxima/minima. This is your SECOND
ORDER CONDITION (SOC) and determines
maxima/minima/inflection point status
3) Obtain the co-ordinates of your
maxima/minima
2.A Local vrs. Global
Thus far, our efforts have revealed LOCAL
maxima and minima.
It is possible, however, that such these
values are not the maximum or minimum
possible.
These values may not be the best policy
decision for a government, individual, or
firm.
3.2 Probabilities
• Probabilities are assigned to the various
outcomes of random variables
Terminology:
Sample Space – set of all possible outcomes from
a random experiment
-ie S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
-ie E = {Pass exam, Fail exam, Fail horribly}
Event – a subset of the sample space
-ie B = {3, 6, 9, 12} ε S
-ie F = {Fail exam, Fail horribly} ε E
3.2 Probabilities
Terminology:
Mutually Exclusive Events – Two events are
mutually exclusive if they cannot occur at the
same time
-ie: rolling both a 3 and an 11; being both dead
and alive; having both a son and a daughter
(and only one child)
Exhaustive Events – cover all possible outcomes
-ie: a dice roll must lie within S ε [2,12]
-ie: a person is either married or not married
3.2 Probability
Probability = measure of likelihood of an event
occurring (between 0 and 1)
P(a) = Prob(a) = probability event a will occur
Prob (Y=y) = probability that the random
variable Y will take on value y
If Prob(a) = 0, the event will certainly never
occur (ie: your instructor turns into a giant
llama)
If Prob(b) = 1, the event will certainly occur (ie:
the sun will rise tomorrow)
3.2 Probability Rules
1) P(a) must be greater than or equal to 0 and
less than or equal to 1 : 0≤P(a) ≤1
2) If a set of events {A,B,C} are exhaustive, then
P(A or B or C) = 1
Ie: Prob. a die roll is between 2 and 12
3) If a set of events {A,B,C} are mutually
exclusive, then
P(A or B or C)=P(A)+P(B)+P(C)
Ie: Prob. of drawing a heart or spade
3.3 Expected Values
Expected Value – measure of central tendency;
center of the distribution; population mean
Discrete Variable:
E(Y) = Σyf(y)
Continuous Variable:
E(Y) = ∫f(y)dy
3.3.1 Properties of Expected Values
a) Constant Property
E(a) = a if a is a constant or non-random variable
Ie: E(14)=14
Ie: E(b1+b2Xi) = b1+b2Xi
b) Constants and non-random variables
E(a+bW) = a+bE(W)
If a and b are non-random and W is random
3.4 Variance Formula
Var(Y) = E(Y-E(Y))2
= E(Y2) – [E(Y)]2
Discrete Random Variable:
Var(Y) = Σ(y-E(Y))2f(y)
Continuous Random Variable:
Var(Y) = ∫(y-E(Y))2f(y)dy
3.4.1 Properties of Variance
a) Constant Property
Var(a) = 0 if a is a constant or non-random
variable
Ie: Var(14)=0
Ie: Var(b1+b2Xi) = 0
b) Constants and non-random variables
Var(a+bW) = b2 Var(W)
If a and b are non-random and W is random
3.4.1 Properties of Variance
c) Covariance Property
If W and V are random variables, and a, b, and c
are non-random, then
Var(a+bW+cV)= Var(bW+cV)
= b2 Var(W) + c2 Var (V)
+2bcCov(W,V)
Where Covariance will be examined in 3.7
3.5 Common Economic Distributions
In order to test assumptions and models,
economists need be familiar with the following
distributions:
 Normal
 t
 Chi-square
 F
Examples and explanations of these tables are
available at
http://www.statsoftinc.com/textbook/sttable.html
3.5 Normal Distribution
The Normal (Z) Distribution produces a bell curve
with a mean of zero and a standard deviation
of one.




The probability that z>0 is always 0.5
The probability that z<0 is always 0.5
Z-tables generally measure area from the centre
Probabilities decrease as you move from the mean of
zero
3.5 Converting to a normal distribution
Z distributions assume that the mean is zero and
the standard deviation is one.
If this is not the case, the distribution needs to
be converted to a normal distribution using
the following formula:
Z = (x-u)/sd
Where
x = value
u = mean
sd= standard deviation
3.5 t-distribution
t-distributions can be 1-tail or 2-tail tests
Interpolation is often needed within the table
Example:
Find the critical t-value (t*) that cuts of 1% of
the right tail with 35df
For 1T=0.01, df 30 gives t*=2.457
df 40 gives t*=2.423
A good approximation of df 35 would be:
t*=(2.457+2.423)/2 = 2.440
3.5 chi-square distribution
Chi-square distributions are 1-tail tests
Interpolation is often needed within the table
Example:
Find the critical chi-squared value that cuts of 5%
of the right tail with 2df
For Right Tail = 0.05, df=2
Critical Chi-Squared Value = 5.99146
3.5 F-distribution
F-distributions are 1-tail tests
Interpolation is often needed within the table
Example:
Find the critical F value (F*) that cuts of 1% of
the right tail with 3df in the numerator and
80df in the denominator
For Right Tail = 0.01, df1=3, df2=80,
df2=60 gives F*=4.13 df2=120 gives F*=3.95
3.5 Interpolation
df2=60 gives F*=4.13 df2=120 gives F*=3.95
Since 80 is 1/3rd of the way between 60 and 120:
60
80
100
120
Our F-value should be 1/3 of the way between
4.13 and 3.95:
4.13
?
3.95
Approximatation:
F*=4.13-(4.13-3.95)/3=4.07
3.6 Joint Probability Density Functions
Joint Probability Density Function-summarizes the probabilities associated
with the outcomes of pairs of random variables
f(p,q) = Prob(P=p and Q=q)
∑ f(p,q) = 1
Similar statements are valid for continuous
random variables.
3.6 Joint and Marginal Pdf’s
Marginal (individual) pdf’s can be determined
from joint pdf’s. Simply add all of the joint
probabilities containing the desired outcome of
one of the variables.
Ie: f(Y=7)=∑f(Y=7,Z=zi)
Probability of Y=7 = sum of ALL joint
probabilities where Y=7
3.6 Conditional Probability Density
Functions
Conditional Probability Density Function-summarizes the probabilities associated
with the possible outcomes of one random
variable conditional on the occurrence of a
specific value of another random variable
Conditional pdf = joint pdf/marginal pdf
Or
Prob(a|b) = Prob(a&b) / Prob(b)
(Probability of “a” GIVEN “b”)
3.6 Statistical Independence
If two random variables (W and V) are
statistically independent (one’s outcome
doesn’t affect the other at all), then
f(w,v)=f(w)f(v)
Therefore:
1) f(w)=f(w|any v)
2) f(v)=f(v|any w)
As seen in the previous example.
3.6 Conditional Expectations and Variance
Assuming that our variables take numerical
values (or can be interpreted numerically),
conditional expectations and variances can be
taken:
E(P|Q=500)=Σpf(p|Q=500)
Var(P|Q=500)=Σ[p-E(P|Q=500)]2f(p|Q=500)
Ie) money spent on a car and resulting utility
(both random variables).
3.7 Discrete and Continuous
Covariance
Discrete Random Variable:
Cov(V ,W )   (v  E (v))( w  E ( w) f (v, w)
v
w
Continuous Random Variable:
Cov(V ,W )    (v  E (v))( w  E ( w) f (v, w)vw
v w
3.7 Correlation
Correlation:
Cov
(
V
,
W
)
Cor (V ,W ) 
Cor (V ,W ) 
sd (V ) sd (W )
 (v  E (v))( w  E (w) f (v, w)
v
w
Var (v) Var ( w)
3.8 Estimators
Population Expected Value:
μ = E(Y) = Σ y f(y)
Sample Mean:
Y

Y
i
N
__
Note: From this point on, Y may be expressed as
Ybar (or any other variable - ie:Xbar). For
example, via email no equation editor is
available, so answers will be in this format.
3.8 Estimators
Population Variance:
σ2 = Var(Y) = Σ [y-E(y)] f(y)
Sample Variance:
S
2
y
(Y  Y )


i
N 1
2
3.8 Estimators
Population Standard Deviation:
σ = (σ2)1/2
Sample Variance:
Sy = (Sy2)1/2
3.8 Estimators
Population Covariance:
Cov(V,W)=∑∑(v-E(v))(w-E(w))f(v,w)
Sample Covariance:
(V  V )(W  W )

Cov(V ,W ) 
i
i
N 1
3.8 Estimators
Population Correlation:
ρvw = corr(V,W)= Cov(V,W)/ σv σw
Sample Correlation:
rvw = corr(V,W)= Cov(V,W)/ Sv Sw
3.8 Estimators
Population Regression Function:
Yi = b1 + b2Xi + єi
Estimated Regression Function:

ˆ
ˆ
Yi  b1  b2 X i
3.8 Estimators
OLS Estimation:
B2hat = ∑(Xi-Xbar)(Yi-Ybar)
---------------------∑(Xi-Xbar)2
B1hat = Ybar – B2hatXbar
(
X

X
)(
Y

Y
)

bˆ 
(X  X )
i
i
2
2
i
bˆ1  Y  bˆ2 X
^
Note: b2 may be expressed as b2hat
3.9.2 Fitted or Predicted Values
From the above we see that often the actual data
points lie above or below the estimated line.
Points on the line give us ESTIMATED y values for each
given x.
The predicted or fitted y values are found using our x
data and our estimated b’s:

ˆ
ˆ
Yi  b1  b2 X i
3.9.3 Estimating Errors or Residuals
The estimated y values (yhat) are rarely equal to their
actual values (y).
The difference is the error term:

ˆ
Ei  Yi  Y
3.9.5 Statistical Properties of OLS
In our model:
Y, the dependent variable, is made up of two
components:
a) b1 + b2Xi – a non-random component that
indicates the effect of X on Y. In this course,
X is non-random.
b) Єi – a random error term representing other
influences on Y.
3.9.5 Statistical Properties of OLS
Error Assumptions:
a) E(єi) = 0; we expect no error; we assume the
model is complete
b) Var(єi) = σ2; the error term has a constant
variance
c) Cov(єi, єj) = 0; error terms from two different
observations are uncorrelated. If the last
error was positive, the next error need not be
negative.
3.9.5 Statistical Properties of OLS
OLS Estimators are Random Variables:
a) Y depends on є and is thus random.
b) B1hat and B2hat depend on Y
c) Therefore they are random
d) All random variables have probability
distributions, expected values, and variances
e) These characteristics give rise to certain OLS
estimator properties.
3.9.5 OLS is BLUE
We use Ordinary Least Squares estimation
because, given certain assumptions, it is
BLUE:
B est
L inear
U nbiased
E stimator
3.10.1 Formula
Given
P{ X  t * s    X  t * s}  1  
We have an upper limit of:
X t *s
And a lower limit of:
Or:
X t *s
CI  X  t * s
3.11 Hypothesis Testing
Testing Consistency of a Hypothesized Parameter:
1) Form a null and an alternate hypothesis.
H0 = null hypothesis = variable is equal to a
number
Ha = alternate hypothesis = variable is not equal
to a number
Ie)
H0: b2=0
Ha: b2≠0
3.11 Hypothesis Testing
Testing Consistency of a Hypothesized Parameter:
2) Collect appropriate sample data
3) Select an acceptable probability (α) of rejecting
a null hypothesis when it is true
-Type one error
-Lower α, more unlikely to find a sample that
rejects the null hypothesis
- α is often 10%, 5%, or 1%
3.11 Hypothesis Testing
Testing Consistency of a Hypothesized Parameter:
4) Construct an appropriate test statistic
-ensure the test statistic can be calculated from
the sample data
-ensure its distribution is appropriate to that being
tested (ie: t-statistic for test for mean)
3.11 Hypothesis Testing
Testing Consistency of a Hypothesized Parameter:
5) Establish (do not) reject regions
-Construct bell curve
-Tails are Reject H0 regions
-Centre is Do not Reject H0 regions
3.11 Hypothesis Testing
Testing Consistency of a Hypothesized Parameter:
6) Compare the test statistic to the critical statistic
-If the test statistic lies in the tails, reject
-If the test statistic doesn’t lie in the tails, do not
reject
-Never Accept
7) Interpret Results
4.1.2 Measuring Goodness of Fit
On average, OLS works well:
 The average of the estimated errors is zero
 The average of the estimated Y’s is always the average of the
observed Y’s
Proof:
4.1.2 Measuring Goodness of Fit
R2 is constructed by dividing the variation of Y
into two parts:
1) Variation in fitted Yhat terms. This is explained
by the model
2) Variation in the estimated errors. This is NOT
explained by the model.
4.2 Hypothesis Testing
So far, we have assumed:
 The error term, єi, is random with
 E(єi)=0; no expected error
 Var(єi)=σ2; constant variance
 Cov(єi, єj)=0; no covariance between
errors
Now we add the assumption that the error term is
normally distributed. Therefore:
 Єi ~ N(0,σ2)
4.2 Hypothesis Testing
If the error is normally distributed, so will be the Y
term (since the randomness of Y depends on
the randomness of the error term). Therefore:
E(Yi) = E(b1+b2Xi+єi)=b1+b2Xi
Var(Yi) = Var(b1+b2Xi+єi)=Var(єi) = σ2
(Given that only Y and є are random, plus our
error term assumptions.)
Therefore:
Yi ~ N(b1+b2Xi, σ2)
4.2 Hypothesis Testing
Since we don’t know σ2, we can estimate it:
This gives us estimates of the variance of our
coefficients:
4.3.1 Deriving a Confidence Interval
Step 1: Recall Distribution
We know that:
(b1hat-b1)/se(b1hat) has a t distribution with N-2
degrees of freedom
(b2hat-b2)/se(b2hat) has a t distribution with N-2
degrees of freedom
This was derived under hypothesis testing using central
limit theorems.
4.3.1 Deriving a Confidence Interval
Step 4: Rearrange for CI:
Thus the 100(1-α)% Confidence Interval is
defined by the range:
By repeatedly calculating Confidence Intervals
using OLS, 100(1- α)% of these CI’s will
contain the true value of the parameter (b1).
4.4 Prediction in Simple Regression
Models
Linear Model:
YPred=b1hat +b2hatX* +0
b1hat estimates b1
b2hat estimates b2
0 estimates the error term
Model evaluated at X*
4.4 Prediction in Simple Regression
Models
Solution:
Since we can estimate σ2/2,
QPred =exp{g1hat + g2hat ln(P*) + σhat2/2}
g1hat estimates b1
g2hat estimates b2
σhat2 estimates σ2
Model evaluated at P*