The long-run relationship between money and prices in Mexico: 1969-2010.

Munich Personal RePEc Archive

The long-run relationship between

money and prices in Mexico: 1969-2010.

Gomez-Ruano, Gerardo

Universidad Iberoamericana

2014

Online at https://mpra.ub.uni-muenchen.de/93647/

MPRA Paper No. 93647, posted 07 May 2019 13:18 UTC

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1 This is an informal presentation, which follows Priestley (1981), Fuller (1996), Brockwell and Davis (1991), and Box et al. (2008).

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#

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# "

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Fig. 1. Sine and cosine functions

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; "

/= 45° " > ?

2A ! @&(

"

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1 7

B = C cos( ) + sin( )

C ( " "

{B }

B = C cos( ) + sin( )

= C cos( ) + sin( )

= 0 ∙ cos( ) + 0 ∙ sin( )

= 0

" ;

cos( − D) = cos( ) cos(D) + sin( ) sin(D) " C = 0 "

B B = (C sin( ) + cos( ))(C sin( + ) + cos( + ))

= C sin( ) sin( + )

+ C (sin( ) cos( + ) + cos( ) sin( + )) + cos( ) cos( + )

= C sin( ) sin( + ) + C (sin( ) cos( + ) + cos( ) sin( + )) + cos( ) cos( + )

= ( sin( ) sin( + ) + ( cos( ) cos( + )

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= ( cos( )

# " 2 = 0

B" ( 2( sin ( ) + cos ( ) = 1

" " 0

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B = C cos(E ) + sin(E )

= C cos(2A ) + sin(2A )

# B ! "

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1 {E_G}_G/ ! " {#_G}_G/ "{H_G}_G/

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2 That is, J = # + KH = $(cos L + K sin L) = $M^NO, where $ = |J| = √# + H , cos L =^Q₃ , and sin L =^R₃ .

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I = . S M^{N TG( )U|V|}

/

S = WX Y XZ 1

2 (# − KH ), ! ≥ 1 0, ! = 0 1

2 \#^{| |}+ KH_{| |}], ! ≤ −1

K K = −1" 4^_( ) "

{S } _/

! {E_G}_G/ {#_G}_G/ {H_G}_G/

` = # + KH <sub>a</sub> = ` = # + K H

: #$(`) = (` − _a)(`<sup>∗</sup>− <sub>a</sub>) " `^∗ 3

`6 " ` = # + KH `^∗ = # − KH

# {S } _/

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= 0 4 sin(0) = 0"

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Fig. 2. An example of the alias effect

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0 ^@

B " ! "

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# " ! E (0, A

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I = . S M^{N TG( )U|V|}

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: {S } _/ "

! E (0, A 0

# "

3 This maximum observable frequency is also known as the Nyquist frequency.

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= ^eM^{N U} `(E)

e

: `(E) `^∗(E) =

`(−E)" `(E) = 0 2 E < 0 "

`" 3

I

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`(E) 0 S_G"

– A A

D I" 7

#$( ) = #$( ₀) = #$ g ^e `(E)

e h = ^e | `(E)|

e

+

i(E) = | `(E)|

i(E) ! E # i(E) "

i(E) = ℎ(E) E" ℎ(E) > ? "ℎ(E₀) E₀ ^/

) 7

{" } > ?

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4 Recall that if a function is differentiable, it means its derivative is continuous. And continuity of a function (∙) implies that for every k 7 0 and ₀, there is a neighborhood of ₀, such that

| ( ) − ( 0)| < k for all in that neighborhood.

13

ℎ(E) =^l_e^m" −A ≤ E ≤ A" ( "

) ! E " 3 3

!

9 ( = 1"

> ? " !

Fig. 3. The power of a white noise process

$

; |#| < 1 " = 0 "

; $ # ! E7

ℎ(E) = _{e( Q nop(U) Q}^lm _m)" −A ≤ E ≤ A" ( "

: # 7 0" !

! 6 # < 0 9

( = 1 # = 0.5 = 0.25

#$() ) = ^lm_Qm =^r_s

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Fig. 4. An example of the power for an AR(1) process

4 > EE" > ?

" 3

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9 " "

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& % "

4 "

4 { _, } { _, }6 "

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, = ^eM^{N U} ` (E)

e

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, = ^eM^{N U} ` (E)

e

(E) ! E { _, } { _, }6

(E) = t \ ` (E), ` (E)]

#$( ` (E)) #$( ` (E)) ^u t F

" " (E)

" 0 "

{ _, } { _, }" ! E # "

! 0 ≤ (E) ≤ 1 # " 0 "

(E) { _, } { _, } ! E

(E) = t ( ` (E), ` (E))

#$( ` (E)) t

B " " "

(E) " 0 " !

` (E) ` (E)

A 0 " " (E) ≠ (E)

%

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5 At the mexican central bank’s website.

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0 $

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6 This bureau is called the Instituto Nacional de Geografía y Estadística (INEGI).

7 The CPI is known in Mexico as the Indice Nacional de Precios al Consumidor (INPC).

17

Fig. 5. CPI monthly log-change

" " 5 5 '($( " I '($(

; " "

$%&% $%.'"

0 G

0 0

$% ' % (

5 "

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Fig. 6. M1a monthly log-change

$% ' % )

/ > ? " " 0

0 "

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Fig. 7. M4a monthly log-change

# " / $ 2 "

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Fig. 8. Log of the power estimate for prices

0 " &

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* $% ' ( $

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Fig. 9. Log of the power estimate for M1a

B " > ? " <

&" @" '

* $% ' ) 9 " /

$ 7 > ? " 3 0 "

&" @" ' /" ' ^-

8 A seasonality around 2.4 months may seem awkward. It is actually not because 2.4 months are one fifth of a year. In other words, this is a harmonic frequency of the yearly frequency. Together, the cycles of length 6, 3, 2.4, and 2 months, help describe common yet not uniform seasonalities.

22

Fig. 10. Log of the power estimate for M4a

A 0 " /

@ /

$% % "

$ " / " D 0

" 7 $

$ , "

0

* & % " ' (

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Fig. 11. Coherency estimate for (M1a, prices)

: $

! '

* & % " ' ) 9 / " "

' 7

( - "

& " $

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Fig. 12. Coherency estimate for (M4a, prices)

B

$%

; " 0

#

"

$% ( ; "

$ 7

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Fig. 13. Gain estimate of prices over M1a

9 ' " > ? ( '6

& " > ? ( -=

$

$% ) # / " > ? ( '

$ 6 > ? $

'

26

Fig. 14. Gain estimate of prices over M4a

2 / $ " >

? # " $

/ "

5

5 '((%

& $ " ' #

# "

5 B)<+

4

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" ! " 3 0 5

; $= " !

$% % " % # "

; " !

> ? !

! "

* $% % " ' ( $

! ! /

@

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Fig. 15. Coherency at frequency zero for (M1a, prices), rolling sample

$%&% $%.& $%%=

$%.& $%-. "

$%-- $%%/ ^%

* $% % " ' ) /

+ 0 $%-- "

9 Strictly speaking, it was a crawling-peg regime. For detailed accounts of mexican economic history see, for example, Moreno-Brid and Bosch (2010), or Kuntz (2010).

29

Fig. 16. Coherency at frequency zero for (M4a, prices), rolling sample

7

"

$%-- 0

$% # "

D 0 ! " !

# " 0 > ?

* $% ( $

7 "

0

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Fig. 17. Gain at frequency zero from prices over M1a, rolling sample

2 " "

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$ # " 7

0 $%-- 0" 0

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Fig. 18. Gain at frequency zero from prices over M4a, rolling sample

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5 4 4 '($$

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10 Benati employs 25-year-width windows. We chose 15-year-width windows, given the smaller span of our sample. Our results are fairly comparable though.

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9 $ " $J ( -J

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Acknowledgements

# 0 < < L < H H

References

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) ?" " =%" @" -$. -=-

5 " 1 '((% " >1 ) # ?"+ , -

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11 By a “lasting deviation” we mean what happens at the zero frequency or the “long-run”. Recall that the zero frequency measure tells us what happens for arbitrarily long cycles.

34

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I " 4 9 '($( "/ 0 1 &

2 " ) < M 4 N ,

1 " ) $%-( " > # O ?".

3 '" .(" =" $((= $($/

5 " D < D 5 '($( " " 2

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Appendix

; H $%-$ ℎ(E) { }

ℎ(E)v = 12A . w(4)x(4)v cos(4E)

(y ) / (y )

35

z "w(4) " x(4)v

4 "

! "

x(4)v = 1z . ( − ̅)( ^|3|− ̅)

y |3|

/

̅ =_y∑^y_/ " { } " " { }

!

" } = 12^$'

w(4) =_~^r_m•^p€•(~)_~ − cos(‚)ƒ" ‚ =^{„e| |}_…† =^{e| |}₀

# { _, } { _, }"

(E) = ‡( (E)) + (ˆ (E)) ℎ (E)ℎ (E) ‰ ^u

(E) " ˆ (E) ! { _, }

{ _, } ! E" ℎ_NN(E) K = 1,2

! E

"

12 See Priestley (1981) and Andrews (1991).

36

v = 1\ (E)(E) v ] + \ˆ (E)v ] ℎ (E)v ℎ (E)v 2

u

ℎv_ŠŠ(E) "

!

v =(E) _se∑^{(y )}_{/ (y )}w(4)‹x (4)v + x (−4)v Œcos(4E)"

ˆ (E)v =_se∑^{(y )}_{/ (y )}w(4)‹x (4)v − x (−4)v Œsin(4E)"

x (4)v =_y∑ \ _, − •••]\ _, − •••]

# #$\ v ] = }(E) 2z (1 − (E) )

9 " { _, } { _, }"

(E) = ( (E)) + (ˆ (E)) ^u ℎ (E)

v =(E) Ž\ (E)v ] + \ˆ (E)v ] • ^u ℎ (E)v

" ! " "

37

# #$\ v ] = }(E) 2z (E) •1 + 1 (E) ‘