The Myth of Chinese Exchange Rate - Portland State Universityweb.pdx.edu/~crkl/BDE/bsts-cny.pdf · The Myth of Chinese Exchange Rate An Application of Bayesian Structural Time Series

The Myth of Chinese Exchange Rate An Application of Bayesian Structural Time Series Analysis

Kuan-Pin Lin

Professor of Economics Portland State University

Policy Changes and Events Since 2015

• 2015-08-11 中国“8.11汇改” • 2015-11-30 人民币获准“入篮” • 2015-12-11 定期公布CFETS人民币汇率指数 • 2016-06-23 英国退欧 • 2016-10-01 人民币正式“入篮” • 2016-10-29 限制购买保险 • 2016-11-03 限制比特币交易 • 2016-11-08 美国大选 • 2016-11-29 限制对外投资 • 2016-12-31 限制海外购房 • 2017-01-13央行对部分银行进行窗口指导 • 2017-02-20 中间价调整 • 2017-04-19 放宽跨境业务限制 • 2017-05-26 中间价调整 • 2017-09-11 降低购售汇成本

Time Series Analysis

• Decomposition

– Trend, Seasonal, and Error Components

• Transformation

– Log, Box-Cox, …

• Smoothing

– Exponential Smoothing

– Structural (State Space) Time Series

• Bayesian Structural Time Series Analysis

Time Series Data Analysis Using R 4

Structural Time Series Model

• Linear Gaussian State Space Model

– Measurement (Observation) Equation

– State Equations

Time Series Data Analysis Using R 5

1 1

1

1 1...

t t t t

t t t

t t t p t

a a b

b b

s s s

2

2

2

~ (0, )

~ (0, )

~ (0, )

t

t

t

N

N

N

2~ (0, )t t t t ty a s N

Structural Time Series Model

• Special Cases

– Non-Seasonal (or Local Linear Trend) Model

– Local Level Model

Time Series Data Analysis Using R 6

1 1

1

t t t

t t t t

t t t

y a

a a b

b b

2

2

2

~ (0, )

~ (0, )

~ (0, )

t

t

t

N

N

N

1

t t t

t t t

y a

a a

2

2

~ (0, )

~ (0, )

t

t

N

N

Structural Time Series Model

• Linear Gaussian State Space Model

– Matrix Representation

Time Series Data Analysis Using R 7

2

1

0 0 0

~ (0, )

~ (0, )

~ ( , )

t t t t

t t t t

y Z N

T R N V

N C P

2

2

2

0 0

, 0 0

0 0

t

t t

t

V

Time Series Data Analysis Using R 8

1 0 1 0 0

1 1 0 0 0

0 1 0 0 0

0 0 1 1 1

0 0 1 0 0

0 0 0 1 0

1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

Z

T

R

1

2

Assuming 4 :

t

t

t t

t

t

p

a

b

s

s

s

Structural Time Series Model

• StrucTS – function (x, type = c("level", "trend", "BSM"),

init = NULL, fixed = NULL, optim.control =

NULL)

• Extension with Regressors

– b may be time varying as bt

– Xt may be high dimensional

Time Series Data Analysis Using R 9

2

1

0 0 0

~ (0, )

~ (0, )

~ ( , )

t t t t t

t t t t

y Z N

T R N V

N C P

b

x

Bayesian Structural Time Series Model

• Basic Bayesian Regression Model

• Variable Selection in Xb

Time Series Data Analysis Using R 10

1 1 1

1 01

0 0

1

i i

i

i i

r

i

subset of with

b b prior guess of with precission matrix

rows and columns of for which

b b b

b

b

2 2 2

2 2

( , ) ( | ) ( )

| ~ ( , ), ~ ,2 2

p p p

df ssN b

b b

b

Bayesian Structural Time Series Model

• Spike and Slab Prior Distribution

Time Series Data Analysis Using R 11

2 2 2

1 2 2 1 1

2

( , , ) ( | , ) ( | ) ( )

::

~ (1 ) | , ~ , ( )

. ., ~ ,2 2

i i

i i

i

i

p p p p

SlabSpike

N b

df sse g

b b

b

Bayesian Structural Time Series Model

• Structural Time Series Model Estimation Using Kalman Filters

• Spike and Slab Regression for Variable Selection

• Bayesian Model Averaging over the Best Models for Forecasting

Time Series Data Analysis Using R 12

Bayesian Structural Time Series Model

• Using Package bsts – function (formula, state.specification,

family = c("gaussian", "logit", "poisson",

"student"), save.state.contributions = TRUE,

save.prediction.errors = TRUE, data,

bma.method = c("SSVS", "ODA"), prior = NULL,

oda.options = list(fallback.probability = 0,

eigenvalue.fudge.factor = 0.01), contrasts =

NULL, na.action = na.pass, niter,

ping = niter/10, timeout.seconds = Inf, seed =

NULL, ...)

Time Series Data Analysis Using R 13

Bayesian Structural Time Series Model

• AddLocalLevel

– function (state.specification = NULL, y,

sigma.prior = NULL, # SdPrior

initial.state.prior = NULL, # NormalPrior

sdy,

initial.y)

Time Series Data Analysis Using R 14

Bayesian Structural Time Series Model

• AddLocalLinearTrend

– function (state.specification = NULL, y,

level.sigma.prior = NULL, # SdPrior

slope.sigma.prior = NULL, # SdPrior

initial.level.prior = NULL, # NormalPrior

initial.slope.prior = NULL, # NormalPrior

sdy, initial.y)

Time Series Data Analysis Using R 15

Bayesian Structural Time Series Model

• AddSeasonal

– function (state.specification, y, nseasons,

season.duration = 1,

sigma.prior = NULL, # SdPrior

initial.state.prior = NULL, # NormalPrior

sdy)

Time Series Data Analysis Using R 16

Bayesian Structural Time Series Model

• SdPrior

– function (sigma.guess, sample.size = 0.01,

initial.value = sigma.guess, fixed = FALSE,

upper.limit = Inf)

• This sets a gamma prior on 1/σ2. Shape (α) = sigma.guess2 ×sample.size/2 Scale (β) = sample.size/2 If specify an upper limit on σ then support will be truncated.

• NormalPrior

– function (mu, sigma, initial.value = mu, fixed

= FALSE)

Time Series Data Analysis Using R 17

BSTS Application

• Counterfactual Inference

– Estimating the causal effect of a designed intervention on a time series: How the response variable would have evolved after the intervention if the intervention had never occurred?

– Random experiments method vs. non-random experimental approach.

18 Time Series Data Analysis Using R

BSTS Application

• Assumptions on Counterfactual Inference

– There is a set of control time series that were themselves not effected by the intervention.

– The relationship between covariates and treated time series, as established during the pre-intervention period, remains stable throughout the post-intervention period.

19 Time Series Data Analysis Using R

BSTS Application

• Bayesian Counterfactual Inference

– A Bayesian structural time series model is used to estimate and predict the counterfactual.

– Priors are part of the model.

– Given a response time series model, performs posterior inference on the counterfactual by computing estimates of the causal effect.

20 Time Series Data Analysis Using R

BSTS Application

• CausalImpact – function (data = NULL,

pre.period = NULL,

post.period = NULL,

model.args = NULL,

bsts.model = NULL,

post.period.response = NULL,

alpha = 0.05)

21 Time Series Data Analysis Using R

BSTS Application

• References

– CausalImpact 1.2.1

– Inferring Causal Impact Using Bayesian Structural Time Series Models

22 Time Series Data Analysis Using R