Fit cross-price demand with NLS (+ robust fallbacks)

Fits a cross-price demand curve using log10-parameterization for numerical stability. The optimizer estimates parameters on the log10 scale where applicable, ensuring positive constraints are naturally satisfied.

Equation forms:

Exponentiated (default): $$y = Q_{alone} \cdot 10^{I \cdot \exp(-\beta \cdot x)}$$
Exponential (fits on log10 response scale): $$\log_{10}(y) = \log_{10}(Q_{alone}) + I \cdot \exp(-\beta \cdot x)$$
Additive (level on $y$): $$y = Q_{alone} + I \cdot \exp(-\beta \cdot x)$$

where $x$ is the alternative product price (or "cross" price) and $y$ is consumption of the target good.

Optimizer parameters (log10 parameterization):

log10_qalone: $\log_{10}(Q_{alone})$ - baseline consumption when the alternative is effectively absent.
I: cross-price interaction intensity; sign and magnitude reflect substitution/complementarity. Unconstrained (can be negative for substitutes).
log10_beta: $\log_{10}(\beta)$ - rate at which cross-price influence decays as $x$ increases.

Natural-scale values are recovered as $Q_{alone} = 10^{log10\_qalone}$ and $\beta = 10^{log10\_beta}$.

The function first attempts a multi-start nonlinear least squares fit (nls.multstart). If that fails—or if explicit start_values are provided—it falls back to minpack.lm::nlsLM. Optionally, it will make a final attempt with nlsr::wrapnlsr. Returns either the fitted model or a structured object with metadata for downstream methods.

Usage

fit_cp_nls(
  data,
  equation = c("exponentiated", "exponential", "additive"),
  x_var = "x",
  y_var = "y",
  start_values = NULL,
  start_vals = lifecycle::deprecated(),
  iter = 100,
  bounds = NULL,
  fallback_to_nlsr = TRUE,
  return_all = TRUE
)

Arguments

data: A data frame with columns for price and consumption. Additional columns are ignored. Input is validated internally.
equation: Character string; model family, one of c("exponentiated", "exponential", "additive"). Default is "exponentiated".
x_var: Character string; name of the price column in data. Default is "x". The column is renamed to "x" internally; see validate_cp_data().
y_var: Character string; name of the consumption column in data. Default is "y". The column is renamed to "y" internally.
start_values: Optional named list of initial values for parameters log10_qalone, I, and log10_beta. If NULL, the function derives plausible ranges from the data and uses multi-start search.
start_vals: Use start_values instead.
iter: Integer; number of random starts for nls.multstart (default 100).
bounds: Deprecated. Log10-parameterized parameters are naturally unbounded. This argument is ignored but retained for backwards compatibility.
fallback_to_nlsr: Logical; if TRUE (default), try nlsr::wrapnlsr when both multi-start NLS and nlsLM fail.
return_all: Logical; if TRUE (default), return a list containing the model and useful metadata. If FALSE, return the bare fitted model object.

Value

If return_all = TRUE (default): a list of class "cp_model_nls":

model: the fitted object from the successful backend.
method: one of "nls_multstart", "nlsLM", or "wrapnlsr".
equation: the model family used.
start_vals: named list of starting values (final used; kept for backward compatibility).
nlsLM_fit, nlsr_fit: fits from later stages (if attempted).
data: the 2-column data frame actually fit.

If return_all = FALSE: the fitted model object from the successful backend.

Details

Start values. When start_values is missing, the function: (1) estimates a reasonable range for log10_qalone from the observed y, (2) estimates log10_beta from the price range, and (3) launches a multi-start grid in nls.multstart.

Zero handling for exponential equation. Since the exponential equation fits on the $\log_{10}(y)$ scale, observations with $y \le 0$ are automatically removed with a warning. Use the exponentiated or additive forms if you need to retain zero consumption values.

Fitting pipeline (short-circuiting):

nls.multstart::nls_multstart() with random starts.
If that fails (or if start_values provided): minpack.lm::nlsLM() using start_values (user or internally estimated).
If that fails and fallback_to_nlsr = TRUE: nlsr::wrapnlsr().

The returned object has class "cp_model_nls" (when return_all = TRUE) with components: model, method (the algorithm used), equation, start_vals, nlsLM_fit, nlsr_fit, and the data used. This is convenient for custom print/summary/plot methods.

Convergence & warnings

Check convergence codes and residual diagnostics from the underlying fit.
Poor scaling or extreme y dispersion can make parameters weakly identified.
For "exponential", the model fits on the $\log_{10}(y)$ scale internally.

Examples

## --- Example: Real data (E-Cigarettes, id = 1) ---
dat <- structure(list(
  id = c(1, 1, 1, 1, 1, 1),
  x = c(2, 4, 8, 16, 32, 64),
  y = c(3, 5, 5, 16, 17, 13),
  target = c("alt", "alt", "alt", "alt", "alt", "alt"),
  group = c("E-Cigarettes", "E-Cigarettes", "E-Cigarettes",
            "E-Cigarettes", "E-Cigarettes", "E-Cigarettes")
), row.names = c(NA, -6L), class = c("tbl_df", "tbl", "data.frame"))

## Fit the default (exponentiated) cross-price form
fit_ecig <- fit_cp_nls(dat, equation = "exponentiated", return_all = TRUE)
summary(fit_ecig)         # model summary
#> Cross-Price Demand Model Summary
#> ================================
#> 
#> Model Specification:
#> Equation type: exponentiated 
#> Functional form: y ~ (10^log10_qalone) * 10^(I * exp(-(10^log10_beta) * x)) 
#> Fitting method: nls_multstart 
#> Method details: Multiple starting values optimization with nls.multstart 
#> 
#> Coefficients:
#>               Estimate Std. Error t value  Pr(>|t|)    
#> log10_qalone  1.194135   0.060311 19.7995 0.0002815 ***
#> I            -1.268376   0.837302 -1.5148 0.2270524    
#> log10_beta   -0.737728   0.265129 -2.7825 0.0688459 .  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Confidence Intervals:
#>               2.5 % 97.5 %
#> log10_qalone  1.002  1.386
#> I            -3.933  1.396
#> log10_beta   -1.581  0.106
#> 
#> Fit Statistics:
#> R-squared: 0.8597 
#> AIC: 34.06 
#> BIC: 33.23 
#> 
#> Parameter Interpretation (natural scale):
#> qalone (Q_alone): 15.64  - consumption at zero alternative price
#> I: -1.268  - interaction parameter (substitution direction)
#> beta: 0.1829  - sensitivity parameter (sensitivity of relation to price)
#> 
#> Optimizer parameters (log10 scale):
#> log10_qalone: 1.194 
#> log10_beta: -0.7377 
fit_ecig$method           # backend actually used (e.g., "nls_multstart")
#> [1] "nls_multstart"
coef(fit_ecig$model)      # parameter estimates: log10_qalone, I, log10_beta
#> log10_qalone            I   log10_beta 
#>    1.1941354   -1.2683758   -0.7377282