# uncomment the next lines, the first time you run
# import Pkg; Pkg.add(["DBnomics", "Dynare", "DataFrames"])
# import Pkg; Pkg.add(url="https://github.com/sdBrinkmann/HPFilter.jl") # we install hpfilter from githubFirst steps with Dynare and moment matching.
Business Cycles and Fluctuations - AE2E6
Objectives
Using Dynare:
- get familiar with the structure of a modfile
- interpret results
- compare moments with the data
- edit a model
- fix steady-state
- extend a model (in particular recheck/rederive first order conditions)
Modfiles
No matching items
Uncomment the following lines to install needed packages (if not already present).
Using Dynare
Download the rbc modfile.
using DynareOpen the RBC model and solve the model. Fix the mistakes in the modfile if any.
context = @dynare "rbc.mod";
# ; mutes the output
# graphs and other outputs are stored in subdirectory `rbc`
# the structure `context contains all the results.`Dynare version: 0.9.10
2024-04-12T11:45:17.545: Starting @dynare rbc.mod
["rbc.mod", "language=julia", "json=compute"]
Dynare preprocessor version: 6.4.0+0
2024-04-12T11:45:17.550: End of preprocessing
2024-04-12T11:45:17.553: Start parse_statements!
2024-04-12T11:45:17.558: End parse_statements!
2024-04-12T11:45:17.558: End parserStarting preprocessing of the model file ...
Found 8 equation(s).
Evaluating expressions...
Computing static model derivatives (order 1).
Normalizing the static model...
Finding the optimal block decomposition of the static model...
2 block(s) found:
1 recursive block(s) and 1 simultaneous block(s).
the largest simultaneous block has 7 equation(s)
and 5 feedback variable(s).
Computing dynamic model derivatives (order 2).
Normalizing the dynamic model...
Finding the optimal block decomposition of the dynamic model...
2 block(s) found:
1 recursive block(s) and 1 simultaneous block(s).
the largest simultaneous block has 7 equation(s)
and 5 feedback variable(s).
JSON written after Computing step.
Preprocessing completed.
longname: c
texname: c
symboltype: Endogenous
orderintype: 6
longname: r
texname: r
symboltype: Endogenous
orderintype: 7
longname: khi
texname: khi
symboltype: Parameter
orderintype: 5
longname: a
texname: a
symboltype: Endogenous
orderintype: 4
longname: delta
texname: delta
symboltype: Parameter
orderintype: 2
longname: nss
texname: nss
symboltype: Parameter
orderintype: 4
longname: y
texname: y
symboltype: Endogenous
orderintype: 3
longname: i
texname: i
symboltype: Endogenous
orderintype: 2
longname: w
texname: w
symboltype: Endogenous
orderintype: 8
longname: k
texname: k
symboltype: Endogenous
orderintype: 1
endogenous_nbr: 8
exogenous_nbr: 1
lagged_exogenous_nbr: 0
exogenous_deterministic_nbr: 0
parameter_nbr: 7
original_endogenous_nbr: 8
lead_lag_incidence: [1 0 0 2 0 0 0 0; 3 4 5 6 7 8 9 10; 0 0 0 0 0 11 12 0]
n_static: 4
n_fwrd: 2
n_bkwrd: 2
n_both: 0
n_states: 2
DErows1: [1, 2, 3, 4]
DErows2: Int64[]
n_dyn: 4
i_static: [2, 3, 5, 8]
i_dyn: [1, 4, 6, 7]
i_bkwrd: [1, 4]
i_bkwrd_b: [1, 4]
i_bkwrd_ns: [1, 2]
i_fwrd: [6, 7]
i_fwrd_b: [6, 7]
i_fwrd_ns: [3, 4]
i_both: Int64[]
i_non_states: [2, 3, 5, 6, 7, 8]
p_static: [4, 5, 7, 10]
p_bkwrd: [1, 2]
p_bkwrd_b: [1, 2]
p_fwrd: [11, 12]
p_fwrd_b: [11, 12]
p_both_b: Int64[]
p_both_f: Int64[]
i_current: [1, 2, 3, 4, 5, 6, 7, 8]
p_current: [3, 4, 5, 6, 7, 8, 9, 10]
n_current: 8
i_current_ns: [1, 2, 3, 4]
p_current_ns: [3, 6, 8, 9]
n_current_ns: 4
icolsD: [1, 2, 3, 4]
jcolsD: [3, 6, 11, 12]
icolsE: [1, 2, 3, 4]
jcolsE: [1, 2, 8, 9]
colsUD: Int64[]
colsUE: Int64[]
i_cur_fwrd: [1, 2]
n_cur_fwrd: 2
p_cur_fwrd: [8, 9]
i_cur_bkwrd: [1, 2]
n_cur_bkwrd: 2
p_cur_bkwrd: [3, 6]
i_cur_both: Int64[]
n_cur_both: 0
p_cur_both: Int64[]
gx_rows: [3, 4]
hx_rows: [1, 2]
i_current_exogenous: [13]
i_lagged_exogenous: Int64[]
serially_correlated_exogenous: Int64[]
Sigma_e: [8.099999999999999e-5;;]
maximum_endo_lag: 1
maximum_endo_lead: 1
maximum_exo_lag: 0
maximum_exo_lead: 0
maximum_exo_det_lag: 0
maximum_exo_det_lead: 0
maximum_lag: 1
maximum_lead: 1
orig_maximum_endo_lag: 1
orig_maximum_endo_lead: 1
orig_maximum_exo_lag: 0
orig_maximum_exo_lead: 0
orig_maximum_exo_det_lag: 0
orig_maximum_exo_det_lead: 0
orig_maximum_lag: 1
orig_maximum_lead: 1
dynamic_indices: [1, 4, 6, 7]
current_dynamic_indices: [1, 4, 6, 7]
forward_indices_d: [3, 4]
backward_indices_d: [1, 2]
current_dynamic_indices_d: [1, 2, 3, 4]
exogenous_indices: [13]
NNZDerivatives: [25, 23, -1]
auxiliary_variables: Dict{String, Any}[]
mcps: Tuple{Int64, Int64, String, String}[]
dynamic_g1_sparse_rowval: [3, 7, 5, 3, 4, 3, 8, 4, 6, 7, 8, 4, 5, 2, 4, 6, 1, 2, 8, 7, 2, 6, 1, 1, 5]
dynamic_g1_sparse_colptr: [1, 3, 3, 3, 4, 4, 4, 4, 4, 6, 8, 12, 14, 17, 20, 21, 23, 23, 23, 23, 23, 23, 24, 25, 25, 26]
dynamic_g2_sparse_indices: [[1, 14, 14], [1, 22, 22], [1, 22, 23], [2, 13, 13], [2, 13, 14], [4, 9, 9], [4, 9, 12], [4, 9, 13], [4, 12, 13], [4, 13, 13], [5, 4, 4], [5, 12, 12], [6, 11, 13], [6, 13, 13], [7, 1, 1], [7, 1, 11]]
static_g1_sparse_rowval: [3, 4, 7, 3, 8, 4, 6, 7, 8, 4, 5, 2, 4, 6, 1, 2, 8, 1, 7, 2, 6]
static_g1_sparse_colptr: [1, 4, 6, 10, 12, 15, 18, 20, 22]
dynamic_tmp_nbr: [3, 4, 0, 0]
static_tmp_nbr: [3, 0, 0, 0]
ids: LinearRationalExpectations.Indices([1, 2, 3, 4, 5, 6, 7, 8], [6, 7], [6, 7], [1, 4], Int64[], [2, 3, 5, 6, 7, 8], [2, 3, 5, 8], [1, 4, 6, 7], [1, 4, 6, 7], [1, 2, 3, 4], [3, 4], [1, 2], [3, 6, 8, 9], [4, 5, 7, 10], [13], 8, (D = [1, 2, 3, 4], jacobian = [3, 6, 11, 12]), (E = [1, 2, 3, 4], jacobian = [1, 2, 8, 9]), Int64[], Int64[])
endval_is_reset: false
has_auxiliary_variables: false
has_calib_smoother: false
has_check: false
has_deterministic_trend: false
has_dynamic_file: true
has_endval: false
has_histval: false
has_histval_file: false
has_initval: false
has_initval_file: false
has_planner_objective: false
has_perfect_foresight_setup: false
has_perfect_foresight_solver: false
has_ramsey_model: false
has_shocks: true
has_static_file: true
has_steadystate_file: true
has_stoch_simul: false
has_trends: false
initval_is_reset: false
modfilepath: rbc
irfs: Dict{Symbol, AxisArrayTables.AxisArrayTable}()
endogenous_steady_state: Float64[]
endogenous_terminal_steady_state: Float64[]
endogenous_linear_trend: Float64[]
endogenous_quadratic_trend: Float64[]
exogenous_steady_state: [0.0]
exogenous_terminal_steady_state: Float64[]
exogenous_linear_trend: Float64[]
exogenous_quadratic_trend: Float64[]
exogenous_det_steady_state: Float64[]
exogenous_det_terminal_steady_state: Float64[]
exogenous_det_linear_trend: Float64[]
exogenous_det_quadratic_trend: Float64[]
trends:
stationary_variables: Bool[0, 0, 0, 0, 0, 0, 0, 0]
estimation: Dynare.EstimationResults(Any[], Any[], Any[], Any[], Matrix{Any}(undef, 0, 0), Matrix{Any}(undef, 0, 0), 0)
filter:
forecast: AxisArrayTables.AxisArrayTable[]
initial_smoother:
linearrationalexpectations: LinearRationalExpectations.LinearRationalExpectationsResults(ComplexF64[], [0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0], [0.0 0.0; 0.0 0.0], [0.0; 0.0;;], [0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0], [0.0; 0.0; 0.0; 0.0; 0.0; 0.0;;], [0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0], [0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0;;], [0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0], Bool[0, 0, 0, 0, 0, 0, 0, 0])
simulations: Simulation[]
smoother:
solution_derivatives: Matrix{Float64}[]
analytical_steadystate_variables: [1, 2, 3, 4, 5, 6, 7, 8]
data:
datafile:
params: [0.985, 0.025, 0.33, 0.33, 1.7112427501331826, 1.0, 0.95]
residuals: [6.1794719649538e-310, 0.0, 6.1794718562609e-310, 0.0, 6.17954108527044e-310, 0.0, 6.17962015701543e-310, 5.0e-324]
dynamic_variables: [6.17961328859545e-310, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 6.17961324669474e-310, 6.17946974505755e-310, 7.1e-322, 2.76236e-318, 7.1e-322]
exogenous_variables: [1.0e-323, 6.1796167183312e-310, 6.17962441485657e-310]
observed_variables: String[]
Sigma_m: Matrix{Float64}(undef, 0, 0)
jacobian: Matrix{Float64}(undef, 0, 0)
qr_jacobian: Matrix{Float64}(undef, 0, 0)
model_has_trend: Bool[0]
histval: Matrix{Union{Missing, Float64}}(undef, 0, 0)
homotopy_setup: @NamedTuple{name::Symbol, type::SymbolType, index::Int64, endvalue::Float64, startvalue::Union{Missing, Float64}}[]
initval_endogenous: Matrix{Union{Missing, Float64}}(undef, 0, 0)
initval_exogenous: Matrix{Union{Missing, Float64}}(undef, 0, 0)
initval_exogenous_deterministic: Matrix{Union{Missing, Float64}}(undef, 0, 0)
endval_endogenous: Matrix{Union{Missing, Float64}}(undef, 0, 0)
endval_exogenous: Matrix{Union{Missing, Float64}}(undef, 0, 0)
endval_exogenous_deterministic: Matrix{Union{Missing, Float64}}(undef, 0, 0)
scenario: Dict{Union{Int64, Dates.UTInstant}, Dict{Union{Int64, Dates.UTInstant}, Dict{Symbol, Pair{Float64, Symbol}}}}()
shocks: Float64[]
perfect_foresight_setup: Dict{String, Any}("periods" => 0, "datafile" => "")
estimated_parameters: Dynare.EstimatedParameters(Union{Int64, Pair{Int64, Int64}}[], Union{Missing, Float64}[], Float64[], Union{Pair{String, String}, String}[], Dynare.EstimatedParameterType[], Float64[], Float64[], Float64[], Float64[], Float64[], Float64[], Distributions.Distribution[])Context(Dict{String, DynareSymbol}("c" => , "r" => , "khi" => , "a" => , "delta" => , "nss" => , "y" => , "i" => , "w" => , "k" => …), Model[], , Results(ModelResults[]), , Dict{Any, Any}())(if needed, download the correct model here )
Inspect the various elements of the solution (decision rule, unconditional moments, simulations).
Interpret the effect of a productivity shock. How does it depend on the productivity?
Bonus: open the same model with Dyno (see link on the course webpage). Compare the results.
Computing moments in the data
Download US time series from the world bank for: real gdp, investment, consumption, hours worked.
using Plots
using DBnomics
using DataFrames
using HPFilter: HP
using Statistics: cor, stdWARNING: Method definition HP(Array{T, 1} where T, Int64) in module HPFilter at /home/pablo/.julia/packages/HPFilter/981TT/src/HPFilterFunc.jl:8 overwritten at /home/pablo/.julia/packages/HPFilter/981TT/src/HPFilterFunc.jl:26.
ERROR: Method overwriting is not permitted during Module precompilation. Use `__precompile__(false)` to opt-out of precompilation.# google OECD
df_ = rdb(ids = [
"OECD/QNA/USA.P5.LNBQRSA.Q", # investment
"OECD/QNA/USA.B1_GS1.LNBQRSA.Q", # GDP
"OECD/QNA/USA.P3.LNBQRSA.Q"
]);# keep only the relevant data, in long format
df_long = df_[!,["Subject","period","value"]]740×3 DataFrame
715 rows omitted
| Row | Subject | period | value |
|---|---|---|---|
| String | Date | Float64 | |
| 1 | Gross capital formation | 1970-01-01 | 2.15688e5 |
| 2 | Gross capital formation | 1970-04-01 | 2.14743e5 |
| 3 | Gross capital formation | 1970-07-01 | 217739.0 |
| 4 | Gross capital formation | 1970-10-01 | 2.08844e5 |
| 5 | Gross capital formation | 1971-01-01 | 2.23069e5 |
| 6 | Gross capital formation | 1971-04-01 | 2.27366e5 |
| 7 | Gross capital formation | 1971-07-01 | 2.29696e5 |
| 8 | Gross capital formation | 1971-10-01 | 2.23591e5 |
| 9 | Gross capital formation | 1972-01-01 | 2.34297e5 |
| 10 | Gross capital formation | 1972-04-01 | 2.45149e5 |
| 11 | Gross capital formation | 1972-07-01 | 2.4584e5 |
| 12 | Gross capital formation | 1972-10-01 | 249080.0 |
| 13 | Gross capital formation | 1973-01-01 | 2.62822e5 |
| ⋮ | ⋮ | ⋮ | ⋮ |
| 729 | Final consumption expenditure | 2021-01-01 | 4.31553e6 |
| 730 | Final consumption expenditure | 2021-04-01 | 4.42287e6 |
| 731 | Final consumption expenditure | 2021-07-01 | 4.44763e6 |
| 732 | Final consumption expenditure | 2021-10-01 | 4.48373e6 |
| 733 | Final consumption expenditure | 2022-01-01 | 4.47788e6 |
| 734 | Final consumption expenditure | 2022-04-01 | 4.49347e6 |
| 735 | Final consumption expenditure | 2022-07-01 | 4.51199e6 |
| 736 | Final consumption expenditure | 2022-10-01 | 4.52916e6 |
| 737 | Final consumption expenditure | 2023-01-01 | 4.5728e6 |
| 738 | Final consumption expenditure | 2023-04-01 | 4.5821e6 |
| 739 | Final consumption expenditure | 2023-07-01 | 4.61965e6 |
| 740 | Final consumption expenditure | 2023-10-01 | 4.65017e6 |
df = unstack(df_long,:period, :Subject,:value);df = dropmissing(sort(df))216×4 DataFrame
191 rows omitted
| Row | period | Gross capital formation | Gross domestic product | Final consumption expenditure |
|---|---|---|---|---|
| Date | Float64 | Float64 | Float64 | |
| 1 | 1970-01-01 | 2.15688e5 | 1.32516e6 | 1.13697e6 |
| 2 | 1970-04-01 | 2.14743e5 | 1.32704e6 | 1.13882e6 |
| 3 | 1970-07-01 | 217739.0 | 1.33927e6 | 1.14738e6 |
| 4 | 1970-10-01 | 2.08844e5 | 1.32492e6 | 1.14562e6 |
| 5 | 1971-01-01 | 2.23069e5 | 1.3609e6 | 1.1627e6 |
| 6 | 1971-04-01 | 2.27366e5 | 1.36826e6 | 1.17093e6 |
| 7 | 1971-07-01 | 2.29696e5 | 1.37952e6 | 1.17765e6 |
| 8 | 1971-10-01 | 2.23591e5 | 1.38276e6 | 1.19219e6 |
| 9 | 1972-01-01 | 2.34297e5 | 1.40816e6 | 1.2072e6 |
| 10 | 1972-04-01 | 2.45149e5 | 1.44012e6 | 1.22563e6 |
| 11 | 1972-07-01 | 2.4584e5 | 1.45371e6 | 1.2364e6 |
| 12 | 1972-10-01 | 249080.0 | 1.47806e6 | 1.25882e6 |
| 13 | 1973-01-01 | 2.62822e5 | 1.51464e6 | 1.2765e6 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
| 205 | 2021-01-01 | 1.14623e6 | 5.24764e6 | 4.31553e6 |
| 206 | 2021-04-01 | 1.12988e6 | 5.32739e6 | 4.42287e6 |
| 207 | 2021-07-01 | 1.16292e6 | 5.37077e6 | 4.44763e6 |
| 208 | 2021-10-01 | 1.22444e6 | 5.4619e6 | 4.48373e6 |
| 209 | 2022-01-01 | 1.239e6 | 5.43472e6 | 4.47788e6 |
| 210 | 2022-04-01 | 1.20802e6 | 5.42704e6 | 4.49347e6 |
| 211 | 2022-07-01 | 1.1905e6 | 5.46278e6 | 4.51199e6 |
| 212 | 2022-10-01 | 1.20503e6 | 5.4975e6 | 4.52916e6 |
| 213 | 2023-01-01 | 1.18421e6 | 5.52808e6 | 4.5728e6 |
| 214 | 2023-04-01 | 1.20309e6 | 5.55634e6 | 4.5821e6 |
| 215 | 2023-07-01 | 1.23273e6 | 5.62267e6 | 4.61965e6 |
| 216 | 2023-10-01 | 1.2424e6 | 5.66821e6 | 4.65017e6 |
rename!(df,[:period, :investment, :gdp, :consumption])216×4 DataFrame
191 rows omitted
| Row | period | investment | gdp | consumption |
|---|---|---|---|---|
| Date | Float64 | Float64 | Float64 | |
| 1 | 1970-01-01 | 2.15688e5 | 1.32516e6 | 1.13697e6 |
| 2 | 1970-04-01 | 2.14743e5 | 1.32704e6 | 1.13882e6 |
| 3 | 1970-07-01 | 217739.0 | 1.33927e6 | 1.14738e6 |
| 4 | 1970-10-01 | 2.08844e5 | 1.32492e6 | 1.14562e6 |
| 5 | 1971-01-01 | 2.23069e5 | 1.3609e6 | 1.1627e6 |
| 6 | 1971-04-01 | 2.27366e5 | 1.36826e6 | 1.17093e6 |
| 7 | 1971-07-01 | 2.29696e5 | 1.37952e6 | 1.17765e6 |
| 8 | 1971-10-01 | 2.23591e5 | 1.38276e6 | 1.19219e6 |
| 9 | 1972-01-01 | 2.34297e5 | 1.40816e6 | 1.2072e6 |
| 10 | 1972-04-01 | 2.45149e5 | 1.44012e6 | 1.22563e6 |
| 11 | 1972-07-01 | 2.4584e5 | 1.45371e6 | 1.2364e6 |
| 12 | 1972-10-01 | 249080.0 | 1.47806e6 | 1.25882e6 |
| 13 | 1973-01-01 | 2.62822e5 | 1.51464e6 | 1.2765e6 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
| 205 | 2021-01-01 | 1.14623e6 | 5.24764e6 | 4.31553e6 |
| 206 | 2021-04-01 | 1.12988e6 | 5.32739e6 | 4.42287e6 |
| 207 | 2021-07-01 | 1.16292e6 | 5.37077e6 | 4.44763e6 |
| 208 | 2021-10-01 | 1.22444e6 | 5.4619e6 | 4.48373e6 |
| 209 | 2022-01-01 | 1.239e6 | 5.43472e6 | 4.47788e6 |
| 210 | 2022-04-01 | 1.20802e6 | 5.42704e6 | 4.49347e6 |
| 211 | 2022-07-01 | 1.1905e6 | 5.46278e6 | 4.51199e6 |
| 212 | 2022-10-01 | 1.20503e6 | 5.4975e6 | 4.52916e6 |
| 213 | 2023-01-01 | 1.18421e6 | 5.52808e6 | 4.5728e6 |
| 214 | 2023-04-01 | 1.20309e6 | 5.55634e6 | 4.5821e6 |
| 215 | 2023-07-01 | 1.23273e6 | 5.62267e6 | 4.61965e6 |
| 216 | 2023-10-01 | 1.2424e6 | 5.66821e6 | 4.65017e6 |
pl1 = plot(df[!,:period], df[!,:investment],label="investment")
pl2 = plot(df[!,:period], df[!,:gdp],label="gdp")
pl3 = plot(df[!,:period], df[!,:consumption],label="consumption")
plot(pl1,pl2,pl3;size=(800,800))Detrend all series.
# detrend the consumption series
series = df[!,:consumption]
trend = HP(series, 1600)
cycle = (series - trend)./trend
df[!,:consumption_trend] = trend
df[!,:consumption_cycle] = cycle;pl1 = plot(series, title="consumption")
plot!(pl1, trend, label="consumption")
pl2 = plot(cycle, title="consumption (cycle)")
plot(pl1, pl2)# detrend gdp
series = df[!,:gdp]
trend = HP(series, 1600)
cycle = (series - trend)./trend
df[!,:gdp_trend] = trend
df[!,:gdp_cycle] = cycle;# detrend the investment series
series = df[!,:investment]
trend = HP(series, 1600)
cycle = (series - trend)./trend
df[!,:investment_trend] = trend
df[!,:investment_cycle] = cycle;Compute the correlations between the detrended series. Compute the ratios \(\frac{\sigma(investment)}{\sigma(gdp)}\) and \(\frac{\sigma (consumption)}{\sigma (gdp)}\). Compare with the RBC model and comment.
# keep only the relevant part of the dataframe
relevant_df = df[:,[:gdp_cycle,:investment_cycle,:consumption_cycle]]216×3 DataFrame
191 rows omitted
| Row | gdp_cycle | investment_cycle | consumption_cycle |
|---|---|---|---|
| Float64 | Float64 | Float64 | |
| 1 | 0.00632641 | -0.00186061 | 0.00492124 |
| 2 | -0.00165798 | -0.0167415 | -0.00174059 |
| 3 | -0.00178395 | -0.0134543 | -0.00245991 |
| 4 | -0.0215434 | -0.063543 | -0.0120762 |
| 5 | -0.00411494 | -0.00995981 | -0.0054254 |
| 6 | -0.00775725 | -0.000994378 | -0.00638908 |
| 7 | -0.00851934 | -0.00060419 | -0.00859888 |
| 8 | -0.0149411 | -0.0363655 | -0.00421424 |
| 9 | -0.00554043 | 0.000597618 | 0.00053205 |
| 10 | 0.00837714 | 0.0378835 | 0.0080933 |
| 11 | 0.00944052 | 0.0323318 | 0.00937495 |
| 12 | 0.0180447 | 0.0380089 | 0.0201799 |
| 13 | 0.0350582 | 0.0876178 | 0.0271306 |
| ⋮ | ⋮ | ⋮ | ⋮ |
| 205 | -0.00632488 | -0.00651634 | -0.00326442 |
| 206 | 0.00311629 | -0.0269902 | 0.0149844 |
| 207 | 0.00553876 | -0.00498233 | 0.0140822 |
| 208 | 0.0167419 | 0.0409745 | 0.0157087 |
| 209 | 0.00586001 | 0.0467197 | 0.00783958 |
| 210 | -0.00135632 | 0.0142257 | 0.00484739 |
| 211 | -0.0005876 | -0.00659147 | 0.00253307 |
| 212 | -5.02405e-5 | -0.000550477 | -5.67176e-5 |
| 213 | -0.000292467 | -0.0237068 | 0.00318781 |
| 214 | -0.000971265 | -0.0140678 | -0.00109743 |
| 215 | 0.00515261 | 0.00421442 | 0.000788311 |
| 216 | 0.00750582 | 0.00610296 | 0.00113667 |
# to compute statistics, we need to convert the dataframe into a matrix
mat = Matrix(relevant_df)using Statistics: std, cov, corcor(mat)3×3 Matrix{Float64}:
1.0 0.909062 0.850059
0.909062 1.0 0.650094
0.850059 0.650094 1.0
std_y = std(relevant_df[:,:gdp_cycle])
std_i = std(relevant_df[:,:investment_cycle])
std_c = std(relevant_df[:,:consumption_cycle])0.010668175144878967println("σ(i)/σ(y): $(std_i/std_y)")
println("σ(c)/σ(y): $(std_c/std_y)")σ(i)/σ(y): 3.3208282679766574
σ(c)/σ(y): 0.6950802322318318Small Open Economy
Start from the same rbc model. Assume the representative agent can save \(b^{\star}_t\) as foreign assets, remunerated at a constant interest rate \(r^{\star}-1\).
Write the new budget constraint for the representative household.
À une date donnée t, le ménage épargne \(b^{\star}_t\) sous forme de titres étrangers et reçoit le remboursement de son épargne de la période précédente \(b^{\star}_{t-1}\) plus les intérêts afférents \((r^{\star} − 1)b^{\star}_{t-1}\) . Par rapport au modèle de base, la contrainte budgétaire en t devient donc :
\[c_t + i_t + b^ {\star}_t \leq w_t n_t + r_t k_{t-1} + r^ {\star} b^{\star}_{t-1}\]
and the lagrangian is:
\[\mathcal{L} = E_t \sum_{j=0}^\infty \beta^j\left[\log c_{t+j} + \chi \dfrac{(1-n_{t+j})^{1-\eta}}{1-\eta} \right.\]
\[+\lambda_{t+j} (w_{t+j}n_{t+j} + r_{t+j}k_{t+j-1} - c_{t+j}-k_{t+j} + (1-\delta)k_{t+j-1} + r^* b^*_{t+j-1} - b^*_{t+j} )\Bigg]\]
Write the new optimality condition.
Il faut ajouter au modèle la condition d’optimalité par rapport à \(b^{\star}_t\) : le ménage représentatif choisit le niveau d’épargne en titres étrangers qui maximise son utilité intertemporelle sous contrainte. On trouve :
\[ -\lambda_t + \beta E_t[\lambda_{t+1}r^*]=0\] soit : \[ \frac{1}{c_t}=\beta r^* E_t\left[\frac{1}{c_{t+1}} \right]\]
What is the long run constraint on interest rate \(r^{\star}\)?
À l’état stationnaire, cette équation implique \(r^{\star}= \frac{1}{\beta}\) . Bien que \(r^{\star}\) représente un taux d’intérêt décidé à l’étranger, \(\frac{1}{\beta}\) est sa seule valeur compatible avec l’existence d’un équilibre de long terme dans le modèle. En effet, il ne peut pas y avoir d’opportunité d’arbitrage à l’équilibre : tous les placements ont la même rentabilité.
Update the modfile (set \(\overline{b^{\star}}=0\))
Pour mettre à jour le fichier .mod, il faut déclarer la nouvelle variable \(b^{\star}\) et le nouveau paramètre \(r^\star\)? en lui donnant la valeur \(\frac{1}{\beta}\) . Il faut ajouter au modèle la condition d’optimalité ci-dessus et mettre à jour l’équation donnant l’équilibre sur le marché des biens. En effet, cette équation est obtenue en équilibre général à partir de la contrainte budgétaire du ménage. En petite économie ouverte, on a :
\[c_t + i_t + b^{\star}_t = w_t n_t + r_t k_{t-1} + r^{\star} b^{\star}_{t-1}\]
Or l’entreprise, supposée en concurrence parfaite, ne fait pas de profit. Donc \(y_t = w_t n_t + r_t k_{t−1}\) et :
\[c_t + i_t + b^{\star}_t - r^{\star} b^{\star}_{t-1} = y_t\]
Ensuite, on remarque que le modèle statique ne détermine pas le niveau de \(b^{\star}\) à l’état stationnaire ; on choisit de le fixer à 0 de sorte que l’état stationnaire des autres variables du modèle ne soit pas affecté. On ajoute donc \(b^{\star}=0\) dans le bloc steady_state_model;.
Print theoretical moments. Comment.
On voit que Dynare ne parvient pas à calculer les moments (moyennes et variances) de certaines variables. De plus, les irfs ne reviennent pas tous à 0.
Simulate the model over 100 periods. Comment.
Avec la commande check;, on voit que le modèle a une valeur propre de module égal à 1 [PAS DANS Dynare Julia]. Dans Dynare, ceci ne provoque pas d’erreur par rapport aux conditions de Blanchard et Kahn mais est problématique : le modèle a une racine unitaire et n’est donc pas stationnaire. En effet, il manque au modèle une force de rappel : si les ménages sont transitoirement incités épargner plus sous forme de titres étrangers, rien ne les incite ensuite de façon endogène à baisser cette épargne vers son niveau initial – la rémunération de cette épargne étant toujours la même.
Assume the foreign interest rate depends on the amount of foreign assets
\[r^{\star}=\frac{1}{\beta} + exp(-\kappa b^{\star}_t) - 1\]
with \(\kappa=0.01\). How do you interpret the equation for \(r^{\star}_t\). How should you modify the model equations?
Plus le ménage détient de titres étrangers, moins ceux-ci sont rémunérateurs. Inversement, plus le ménage est endetté vis-à-vis de l’étranger \((b^{\star} < 0)\), plus les prêteurs étrangers vont exiger un taux d’ intérêt élevé. On peut donc interpréter le terme \(exp(−\kappa b^{\star}_t ) − 1\) comme une prime de risque sur le taux d’intérêt étranger. Pour mettre à jour le fichier .mod, il faut déclarer à présent \(r^{\star}\) comme une variable et non un paramètre, déclarer sa valeur à l’état stationnaire \(r^{\star} = \frac{1}{β}\) dans le bloc steady_state_model; et enfin ajouter l’équation ci-dessus dans le bloc model;.
Si on ne change pas les conditions d’ Euler, cela revient à supposer que les consommateurs/le gouvernment n’internalisent pas l’ effet de leur emprunt sur le taux d’ intérêt.
Update the modfile and comment on the generated moments. Do they depend on the choice of \(\kappa\)?